Molecular

Fullerides

by

Wilfred Kelsham Fullagar

A thesis submitted for the degree of Doctorate of Philosophy of The Australian National University

April, 1997 Declaration

The research described in this dissertation was undertaken primarily at the Australian National University, under the supervision of Professor John White. Data collection described herein was in three instances performed elsewhere: X-ray synchrotron diffraction data was collected at the Australian National Beamline Facility in Tsukuba, Japan; neutron diffraction patterns were recorded at the Intense Pulsed Neutron Source at the Argonne National Laboratories in Illinois, USA; and inelastic neutron scattering measurements were made at the ISIS Pulsed Neutron Source of the Rutherford Appleton Laboratory, Chilton, UK.

Except where otherwise stated, sample preparation and analysis was in all instances performed by the candidate. Note, however, the particularly valuable contributions of Dr. Philip Reynolds and Dr. Lucjan Dubicki in § 4 and § 5, respectively (see Acknowledgments). Funding restrictions prevented the candidate from attending one of the three data collection trips to the Australian National Beamline Facility in Japan, and the relevant diffraction data were measured on his behalf by Dr. Philippe Espeau and Dr. John Watson. Similar restrictions led to the Na2C60, Rb4C60 and Rb3C60(ND3)2.5 TFXA data in § 6 being collected on the candidate's behalf by Dr. Philip Reynolds and Dr. Tony Brown, this being the second of two expeditions to the Rutherford Appleton Laboratory, UK. The associated sample preparations and subsequent data interpretation were by the candidate.

None of the work presented in this thesis has been submitted to any other institution for any degree.

Wilfred K. Fullagar 28/4/1997 Abstract

The closed shell structures of certain all-carbon fragments originally observed in mass spectroscopy experiments leads to the enhanced stability of these species, known as fullerenes, which have excited sufficient interest amongst chemists and physicists over the last decade to warrant the award of the 1996 Nobel Prize for Chemistry to their discoverers.

Studies of the stability, symmetry, and consequent remarkable properties of fullerenes began in earnest in 1991 with the development of a technique enabling the production and purification of macroscopic quantities of material. The best known and most widely studied fullerene is the truncated icosahedral C60 molecule, which forms the basis of the present work.

One important property of C60 is that it forms salts with sufficiently electropositive species, such as the alkali metals. The resulting salts contain C60 anions and are known as fullerides. Certain of these salts display metallic behaviour, and some superconduct at temperatures as high as 33 K.

Three aspects of fulleride research are addressed in this work. These are: i) the preparation, crystal structure determination and superconductivity characterization of several new fullerides, particularly those including ammonia as an additional intercalant,

n- ii) the electronic structure of the C60 (n = 1 - 6) anions, as probed by solution-phase near infrared absorption spectroscopy, and iii) the molecular dynamics of a number of fullerides, superconducting and non- superconducting, by inelastic neutron scattering.

This work has grown out of an Honours project also concerning C60, the combined duration of the two studies covering essentially the entire history of this widely and competitively studied field. Acknowledgments

I owe a great deal to my supervisors, Professor John White, Dr. Graham Heath and Dr. Richard Bramley, for their encouragement and patient enthusiasm throughout my time at the Research School of Chemistry. Professor White is owed particular personal thanks for accepting me as a PhD student in perhaps rather trying circumstances!

The contributions of Dr. Philip Reynolds to this work, and to four of the five publications that have arisen from it are nothing short of vital. He introduced me to crystallographic structure refinement techniques (§ 4), and certain of the refinements described in this work were initiated and indeed completed by him. His understanding of neutron scattering has also been extremely valuable. Patience and clarity as a teacher, a deep understanding of general physical chemistry, and an approachable and freindly manner are qualities of his that have been appreciated by all members of the White group, and his input to the latter cannot be understated.

Dr. Lucjan Dubicki is another key contributor, and was kind enough to embrace the very challenging problem of the electronic structure in the near-infrared absorption spectra of the

C60 anions (§ 5), with the associated wealth of chemical literature, all at extremely short notice. The present understanding of the near infrared absorption spectra as arising from a combination of interelectron repulsion and Jahn-Teller effects is in large measure his achievement, and it is regretted that he was not approached on the topic much sooner.

The technical staff at the Research School of Chemistry played a crucial role in this work for their construction and assistance in the design of many and varied pieces of apparatus, ranging from hermetically sealable cryogenic sample cans to the liquid ammonia titrator described in § 3.1.3; Chris Tomkins' input in the design and construction of the latter having been particularly noteworthy. Particular thanks are due to Mr Gordon Lockhart, whose assistance and expertise in the laboratory on innumerable occasions proved him quite indispensable. I would especially like to thank him for teaching me the rudiments of glassblowing, a skill which enabled a high degree of independence in the construction of laboratory apparatus.

I thank the Australian Nuclear Science and Technology Organization (ANSTO) for making possible access to the international facilities described in § 8, as well as providing the necessary funding.

Finally, this work would have been quite impossible had it not been for the constancy of parents, siblings and friends, whose apparently endless compassion and forbearance in the face of all manner of grievances have always been and continue to be a source of particular strength. Table of Contents

1. SCOPE OF THESIS ...... 1-1

1.1 REFERENCES...... 1-4

2. STRUCTURES AND SUPERCONDUCTING PROPERTIES ...... 2-1

2.1 BASIC STRUCTURAL MOTIFS...... 2-1 2.1.1 Face-Centred Cubic Structures ...... 2-1 2.1.2 Body-Centred Structures...... 2-6 2.1.3 Other Packing Schemes...... 2-8

2.1.4 C60 Molecular Structure Changes ...... 2-9 2.1.5 Oligomeric phases ...... 2-9

2.2 SOLID-STATE BAND STRUCTURES...... 2-11

2.3 FULLERIDE SUPERCONDUCTIVITY REVIEW...... 2-18 2.3.1 Relationship between structure and superconductivity...... 2-19 2.3.2 Isotope effects...... 2-21 2.3.3 Superconducting Parameters of the Fullerides ...... 2-22 2.3.4 Electronic Pairing Model of Fulleride Superconductivity...... 2-22 2.3.5 Electron-Phonon Coupling Model of Fulleride Superconductivity...... 2-24

2.4 REFERENCES...... 2-28

3. SYNTHESIS AND SUPERCONDUCTIVITY...... 3-1

3.1 PREPARATIVE METHODS...... 3-1 3.1.1 Vapour-phase intercalation ...... 3-1 3.1.2 Preparation via decomposition of alkali-containing compounds...... 3-3 3.1.3 Preparation by titration...... 3-4 3.1.4 Bomb synthesis ...... 3-10 3.1.5 Preparation in Organic Solvents...... 3-12 3.1.6 Electrochemical Intercalation and Electrocrystallization...... 3-12

3.2 SUPERCONDUCTIVITY MEASUREMENTS...... 3-13 3.2.1 Low-Field Microwave Absorption...... 3-13 3.2.2 Radio Frequency Susceptibility Measurements...... 3-18

3.3 REFERENCES...... 3-22 4. FULLERIDE STRUCTURE DETERMINATION ...... 4-1

4.1 FULLERIDE AMMONIATES...... 4-2

4.1.1 Li3C60(NH3)4 ...... 4-2

4.1.2 Na3C60(NH3)6...... 4-3

4.1.3 Li~2C60(NH3)~8...... 4-7

4.1.4 Na2C60(NH3)8...... 4-9

4.1.5 RbxC60(NH3)y...... 4-11

4.2 FULLERIDES PREPARED BY DEAMMONIATION ...... 4-16 4.2.1 Lithium and Sodium Fullerides...... 4-16 4.2.2 Rubidium Fullerides...... 4-19

4.3 NA2C60 AND RB4C60 TFXA SAMPLES...... 4-20

4.3.1 Na2C60...... 4-20

4.3.2 Rb4C60...... 4-24

4.4 REFERENCES...... 4-27

5. MOLECULAR ELECTRONICS ...... 5-1

5.1 BACKGROUND...... 5-1

5.2 EXPERIMENTAL RESULTS...... 5-3

5.3 VIBRONIC STRUCTURE ...... 5-11

5.4 NEAR-INFRARED ELECTRONIC TRANSITIONS...... 5-14

5.5 REFERENCES...... 5-25

6. DYNAMICS...... 6-1

6.1 INTERMOLECULAR DYNAMICS...... 6-1

6.1.1 C60 Intermolecular Dynamics...... 6-1 6.1.2 Alkali Fulleride Intermolecular Dynamics...... 6-3 6.1.3 TFXA Samples and Sample Cell Contributions...... 6-5 6.1.4 TFXA Intermolecular Inelastic Neutron Scattering...... 6-8

6.2 INTRAMOLECULAR VIBRATIONAL SPECTRA...... 6-11 6.2.1 Comparison with Optical Spectra ...... 6-11 6.2.2 TFXA Intramolecular Fulleride Data ...... 6-21 6.2.3 Experimental Considerations...... 6-23 6.2.4 TFXA data between 200 and 450 cm-1...... 6-27 6.2.5 TFXA data between 450 and 600 cm-1...... 6-35 6.2.6 TFXA data between 600 and 900 cm-1...... 6-36 6.2.7 TFXA data above 900 cm-1...... 6-37

6.3 REFERENCES...... 6-42 7. CONCLUSIONS ...... 7-1

8. APPENDICES ...... 8-1

8.1 SUPERCONDUCTING PROPERTIES...... 8-1

8.2 OVERSEAS FACILITIES...... 8-4 8.2.1 The General Purpose Powder Diffractometer (GPPD) ...... 8-7 8.2.2 The Time Focused Crystal Analyser (TFXA) Spectrometer ...... 8-8 8.2.3 The Australian National Beamline Facility (ANBF):...... 8-11

8.3 REFERENCES...... 8-13 1. Scope of Thesis

The fullerenes are a family of carbon allotropes characterized by closed shells of carbon atoms. A few typical members of the family are shown in Figure 1.1.

Figure 1.1: Several typical fullerenes. Clockwise from top left: C60 (Ih), C76 (D2), C78 (C2v) and C84 (D2). These images were downloaded from a publicly accessible internet address1.

Fullerene research began in 1985 when a group interested in the nucleation of carbon from red giant stars performed experiments in which a mass spectrometer was used to detect fragments produced by the laser ablation of graphite. Two molecular fragments

+ + were found to be particularly stable, corresponding to C60 and C70 . The stability of the

+ 2 C60 fragment was attributed to the unstrained and symmetrical structure illustrated .

1-1 The very high symmetry of the C60 fragment was quick to attract the attention of theoreticians3 4 5 6 7 8 9 10 11 12, but it was not until late 1990 that the synthesis of milligram quantities of the material13 caused the sudden "epidemic"14 spread of fullerene research, with papers appearing in the literature at the prodigious rate of several per day! A growing number of books15 16 17 18 19 20 21 22 23 and journal reviews24 25 26 27 28 29 30 31 32 33 34 have been published on the general topic of fullerenes and their derivatives. The journal Fullerene Science and Technology35 also attracts many research articles. Figure 1.1 and one or two other images in this thesis were downloaded from publicly accessible internet addresses; in fact the internet is also an important source of information on the very rapidly developing fullerene field36.

This project focuses almost exclusively on the physical and chemical properties of the compounds of C60, the most symmetrical and abundant fullerene produced in the electric 13 arc synthesis . The aim has been to devise a ready synthesis for salts of C60 (known as fullerides), to define their structure and superconductivity, and to investigate the superconducting mechanism.

In all known fullerides, the C60 is present as an anion. The best-known of these compounds contain alkali or alkaline-earths as counter-cations, but other salts incorporating lanthanides and organic cations are also known. Certain of these salts are 37 superconducting , some having transition temperatures (Tcs) surpassed only by the cuprate materials. A positive correlation between the fulleride lattice parameter (which 38 is governed essentially by the size of the alkali metal ion) and Tc was observed , with structural transitions and electronic instabilities ultimately placing an upper limit on the experimentally observed values. The superconductivity and electronic structure in the fullerides is also sensitive to the C60 orientational ordering.

The intercalation of small inert molecules can be anticipated to increase lattice parameters and therefore possibly also the superconducting transition temperatures. It has been found that the Tc of Na2CsC60 is increased from 10.5 K to 29.6 K upon 39 intercalation of gaseous ammonia, with formation of the compound Na2CsC60(NH3)4 . Much of the work presented in this thesis is related to the synthesis and characterization of fullerides that have been prepared in liquid ammonia, which proves to be an extremely versatile solvent for such preparations. Although we have not observed

1-2 superconductivity in any of the new compounds described in this thesis, a variety of new structures have been discovered, and an ammonia-induced disproportionation is reported.

One of the synthetic approaches employed in this work involved the spectrophotometric titration of a suspension of C60 with a solution of alkali metal in liquid ammonia. In this

2- 3- 4- way it was possible to obtain the clearest C60, C60 and C60 absorption spectra yet

5- 6- observed, as well as the first absorption spectra of C60 and C60. Although the electronic structures of the C60 anions remain an area of active research, we attempt the interpretation of these spectra in terms of competition between on-ball interelectronic

n- repulsion and the Jahn-Teller effect (which operates in the C60 (n = 1 - 5) anions), as well as superimposed vibronic structure. The vibrational properties of C60 anions are of particular interest in the context of superconductivity, and led to efforts to enhance the vibronic fine structure that was observed in the titration experiments.

There is a growing body of evidence to suggest that certain of the intramolecular vibrational modes of the C60 molecule play a fundamental role in fulleride superconductivity. As part of this project, a series of simple fulleride salts has been investigated by inelastic neutron scattering, a technique which measures the whole density of states for the lattice and molecular modes, thus yielding information unobtainable by the more conventional Raman and infrared spectroscopies. These experiments add experimental constraints to the interpretation of the lattice dynamics, and show evidence of doping-induced dispersion among the intramolecular modes. Knowledge of such dispersion is invaluable in the interpretation of spectral line- broadening in the fullerides, which has also been related to the electron-phonon interactions believed responsible for superconductivity.

Another model exists for fulleride superconductivity in which electronic correlation effects in certain C60 anions can lead to an effective attraction between valence electrons and consequent formation of superconducting electron pairs. The two models and their relative merits are discussed in terms of the available experimental data.

1-3 1.1 References

1 World Wide Web site: "http://www.susx.ac.uk/Users/kroto/fullerenes.html" 2 H.W. Kroto, J.R. Heath, S.C. O'Brien, R.F. Curl and R.E. Smalley, Nature, 318, 162 (1985) 3 R.C. Haddon, L.E. Brus, K. Raghavachari, Chem. Phys. Lett., 125, 459 (1986) 4 P.W. Fowler and J. Woolrich, Chem. Phys. Lett., 127, 78 (1986) 5 R.L. Disch and J. Schulman, Chem. Phys. Lett., 125, 465 (1986) 6 P.D. Hale, J. Am. Chem. Soc., 108, 6087 (1986) 7 Z.C. Wu, D.A. Jelski and T.F. George, Chem. Phys. Lett., 137, 291 (1987) 8 R.E. Stanton and M.D. Newton, J. Phys. Chem., 92, 2141 (1988) 9 F. Negri, G. Orlandi and F. Zerbetto, Chem. Phys. Lett., 144, 31 (1988) 10 W.G. Harter and D.E. Weeks, J. Chem. Phys., 90, 4727 (1989) 11 W.G. Harter and D.E. Weeks, J. Chem. Phys., 90, 4744 (1989) 12 Z. Slanina, J. Rudzinski, M. Togasi and E. Osawa, J. Molec. Struct. (Theochem.), 202, 169 (1989) 13 W. Krätschmer, L.D. Lamb, K. Fostiropoulos and D.R. Huffman, Nature, 347, 354, (1990) 14 T. Braun, Angew. Chem., Int. Ed. Engl., 31, 588 (1992) 15 Fullerenes: Synthesis, Properties and Chemistry of Large Carbon Clusters¸ by G.S. Hammond and V.J. Kuck, ©1992 American Chemical Society, Washington DC. 16 Electronic Properties of Fullerenes: Proceedings of the International Winderchool on Electronic Properties of Novel Materials, Kirchberg, Tirol, March 6-13 1993, by H. Kuzmany, ©1993 Springer Verlag, Berlin. 17 Buckminsterfullerenes, by W.E. Billups and M.A. Ciufolini, ©1993 VCH, New York, USA 18 Handbook of Carbon, Graphite, Diamond and Fullerenes: Properties, Processing, and Applications, by H.O. Pierson, ©1993 Noyes Publications, Park Ridge, N.J. USA. 19 The Fullerenes, by H.W. Kroto, J.E. Fischer and D. Cox, ©1993 Pergamon Press, Oxford UK. 20 Proceedings of the Symposium on Recent Advances in the Chemistry and Physics of Fullerenes and Related Materials (1994: San Fransisco, California), by K.M. Kadish, ©1994 Electrochemical Society, Pennington, N.J. USA. 21 Physics and Chemistry of the Fullerenes, by K. Prassides, ©1994 Kluwer Academic Publishers, Boston, USA 22 The Chemistry of Fullerenes¸ by R. Taylor, ©1995 World Scientific, Singapore. 23 Electronic Structrue Calculations on Fullerenes and their Derivatives, by J. Cioslowski, ©1995 Oxford University Press, Oxford UK. 24 H.W. Kroto, A.W. Allaf and S.P. Balm, Chem. Rev., 91, 1213 (1991) 25 J. Phys. Chem. Solids, 53 (11) (1992) 26 Carbon, 30 (8) (1992) 27 Acc. Chem. Res., 25, (3) (1992) 28 J. Phys. Chem. Solids, 54 (12) (1993) 29 Mat. Sci. and Eng. B, 19 (1-2) (1993) 30 Philosophical Transactions of the Royal Society: Phys. Sci. and Eng., 345 (1667) (1993) 31 Synthetic Metals, 56, pp 2429-3269 (1993) 32 Solid State Physics, 48 (1994)

1-4 33 Synthetic Metals, 64, pp 309-368 (1995) 34 Synthetic Metals, 70, pp 1309-1528 (1995) 35 Fullerene Science and Technology, ISSN 1064-122X, Marcel Dekker Inc., NY 36 Several contents alert facilities exist at the time of writing which may be found at the World Wide Web address "http://www.physik.uni-oldenburg.de/bucky/htmls/bucky.html", and by contacting the e-mail addresses "[email protected]" and "[email protected]". The service provided by "[email protected]" may also be used to search for recent articles on a particular topic. 37 A.F. Hebard, M.J. Rosseinsky, R.C. Haddon, D.W. Murphy, S.H. Glarum, T.T.M. Palstra, A.P. Ramirez and A.R. Kortan, Nature, 350, 600 (1991) 38 R.M. Fleming, A.P. Ramirez, M.J. Rosseinsky, D.W. Murphy, R.C. Haddon, S.M. Zahurak and A.V. Makhija, Nature, 352, 787 (1991) 39 O. Zhou, R.M. Fleming, D.W. Murphy, M.J. Rosseinsky, A.P. Ramirez, R.B. van Dover and R.C. Haddon, Nature, 362, 433 (1993)

1-5 2. Structures and Superconducting Properties

In this chapter we review the current understanding of the fullerides from a structural, electronic, and superconducting point of view. Almost the entirety of this body of knowledge has been established concurrently with the present project.

2.1 Basic Structural Motifs

2.1.1 Face-Centred Cubic Structures

The typical fulleride structure is that adopted by the well-known compounds K3C60 and

Rb3C60, a simplified version of which is shown in Figure 2.1 and which was first described by Stephens et al.1.

Figure 2.1: Merohedrally ordered (see text) representation of the structure adopted by the A3C60 (A = K or Rb) fullerides. Green and blue balls represent alkali metals in tetrahedral and octahedral interstices, respectively.

2-1 Ignoring for the moment the orientation of the C60 balls, the structure shown can be described as face-centred cubic (fcc, space group Fm 3m) with alkali metal ions in the octahedral and tetrahedral interstices between the balls. It is possible to synthesize fullerides with this structure which contain mixtures of alkali metals; in such compounds the smaller alkali metal is generally found in the smaller tetrahedral holes.

Molecular orientation complicates this picture and leads to the differences between what are essentially fcc close-packed structures. The Fm 3m space group requires a four-fold symmetry axis, which the C60 molecule, with Ih symmetry, does not have. In Rietveld fits to powder diffraction patterns, three of the C60 molecular two-fold axes are aligned along the crystallographic axes as shown in Figure 2.1; half the molecules are then rotated by 90° about one of the four-fold crystallographic axes to generate a new molecule centred on the same position. Since the two molecules are related by reflection through a (110) mirror plane, the C60 molecules are said to be merohedrally disordered. This arrangement results in the 1/4,1/4,1/4 (tetrahedral) sites lying directly above hexagonal faces of the C60 molecule, so that the alkali ions in this site have 24 nearest carbon neighbours. The 1/2,1/2,1/2 (octahedral) sites lie along the molecular two-fold axes so that each octahedral alkali ion has 12 carbon nearest neighbours.

Determination of the ball orientation in fullerene-derived compounds is one of the more detailed problems in the structural analysis of these materials which we will generally not explore in detail, since in more complex materials the available powder diffraction data is frequently insufficient to enable such modelling.

The approximation of the C60 molecular structure as spherical is intuitive, and was borne out by the first room-temperature X-ray diffraction patterns, which indicated an fcc arrangement of spherical shells. This can arise from either static or dynamic disorder of the molecules. Subsequent experiments showed that the disorder was due to the rapid spinning of individual molecules2.

On cooling to 258 K C60 undergoes a first-order phase transition which is associated with orientational ordering3, which may also be brought about at room temperature by the application of ~4 GPa pressure4. Below the transition temperature, additional peaks appear in the diffraction pattern which can be indexed as primitive cubic, and fitted

2-2 using the space group Pa 35. The predominant molecular orientation required by low- temperature X-ray and neutron diffraction data can be generated from a merohedrally ordered array of C60s by rotating each of the four molecules in the face-centred unit cell through an angle Γ = 22º about a different one of each of the four crystallographic [111] axes, as shown in Figure 2.2.

Figure 2.2: Generation of the low temperature Pa 3 C60 structure from a merohedrally ordered arrangement involves rotation about each of the illustrated (111) axes by 22º (from Copley et al.6)

Rotation through the angle Γ = 22º corresponds to a global minimum of the potential energy as a function of angle, but there is another local minimum at Γ = 82º, as shown in the upper panel of Figure 2.3. The global minimum corresponds to a situation in which the electron rich bonds between hexagons (6:6 bonds) are adjacent to the relatively electron-deficient pentagons on neighbouring molecules, while in the local minimum the 6:6 bonds are adjacent to neighbouring balls' hexagonal faces. Temperature dependent diffraction data indicate that the energy difference between the two minima in undoped C60 is ~11 meV, while a variety of other measurements indicate an activation energy in the range 180 - 270 meV6. The transition appears to involve rotations around molecular two-fold axes7 rather than the three-fold axes as suggested by Figure 2.2.

2-3 Figure 2.3: Potential energy vs rotation angle Γ (see text) for (top) C60 (using atom-centred Lennard-Jones potentials and bond-centred Coulombic interactions between molecules), (middle) the repulsive A-C60 interaction due to core orbital overlap expected for K3C60, and (bottom) the attractive 8 Coulombic interaction expected for Na2C60. (From Fischer et al. )

The potential energy curve changes upon intercalation of alkali metals, the effects of which may be seen in the lower two panels of Figure 2.3. Repulsive A-C60 interactions become important for the alkali metals in the smaller tetrahedral holes; in the case of undoped C60, with a = 14.2 Å, the van der Waals radii of the tetrahedral and octahedral sites are ~1.1 Å and ~2.1 Å respectively assuming spherical C60 molecules (the tetrahedral site in particular is in fact slightly larger because the C60s are oriented with hexagonal faces over these sites). These hole sizes may be compared with the alkali metals' ionic radii: Li (0.76 Å), Na (1.02 Å), K (1.38 Å), Rb (1.52 Å) and Cs (1.67 Å).

For large alkali metals, in which the repulsive A-C60 forces are dominant, inspection of

Figure 2.3 leads us to expect merohedral disorder of the C60 molecules, while for smaller alkali metals, the Coulombic interactions control the balls' orientation and ordering with Γ ≈ 22º may be anticipated. The orientation is thus governed by the content of the tetrahedral sites.

2-4 The fcc positioning of the C60 balls is preserved even for alkali metals as large as Rb, some of the additional room required in the tetrahedral site being produced by an overall expansion of the lattice to a = 14.43 Å.

While little is known about lithium-containing fullerides, the fcc-derived Li2CsC60 structure (in which the Li atoms occupy tetrahedral sites) differs from its sodium analogue in showing little orientational ordering of the anions at room temperature, 9 despite an asserted propensity for Li-C60 bond formation . It should be noted that the fcc alkali fullerides containing Li and Na in tetrahedral sites actually have smaller lattice parameters than undoped C60, and that for sufficiently small alkali metals, eg. in 10 Li2KC60, the fcc structure appears to be unstable .

Several experimental results have complicated this picture. In the case of K3C60 and

Rb3C60, there are theoretical grounds to suppose that there is a degree of correlation between the orientation of adjacent balls, relating them through a 90° molecular 11 rotation . Although the lack of supercell peaks in the C60 powder patterns shows that this does not extend to long range, neutron diffraction work involving the pair density distribution function in Rb3C60 indicates that there is short-range "antiferromagnetic" ordering12; in other words adjacent balls tend to have opposite orientations.

A challenge to the completeness of the structure as described above is provided by the solid state alkali-metal NMR results originally reported by Walstedt et al.13. Signals corresponding to the alkali metals unexpectedly show three peaks where two were expected (one for ions in octahedral sites, another for the tetrahedral ions). Two of the three peaks are known to correspond to the tetrahedral site, but the cause of the splitting is still open to question. It seems improbable that a C60 molecular distortion such as a Jahn-Teller distortion (see § 5.4) would be sufficient to produce the observed splitting. It has been proposed that tetrahedral site vacancies could be responsible for the NMR spectra, with the temperature dependence of the vacancy mobility accounting for the temperature dependence of the spectra8 14. Such suppositions find some support in very careful refinements of synchrotron diffraction patterns of Rb3C60 at room temperature and 20 K, which have provided evidence for a low concentration (~3.5%) of tetrahedral site vacancies in some samples8.

2-5 EXAFS experiments15 lend weight to the idea that off-centre site occupancy may occur 16 in the large octahedral holes for the classical fcc A3C60 compounds , though detailed X-ray diffraction pattern refinements at both room temperature and 20 K show only isotropic displacement on this site8.

Other complications to the fcc close packing scheme also involve the ions in the octahedral holes. For example, supercell formation due to cation vacancy ordering has 17 been observed in the basically fcc rare earth fulleride Yb2.75C60 , and there is evidence for the ordering of multiple Ca atoms in the octahedral sites of the superconductor 18 Ca5C60 such that the unit cell symmetry changes from fcc to primitive cubic . The sodium fullerides are also unusual in that they retain the fcc structure throughout the stoichiometry range NaxC60, 0 < x < 11, again with the formation of Na clusters in the 19 octahedral site for x > 3. The unusual electronic behaviour of Na3C60, which becomes increasingly metallic at elevated temperatures, is also likely to be related to octahedral 20 21 site disorder . Contrary to expectations, Na3C60 (Pa 3 structure with a = 14.191 Å ) 22 does not superconduct , and this is believed to be due to disproportionation to Na2C60 21 and Na6C60 at sub-ambient temperatures .

2.1.2 Body-Centred Structures

If the alkali metal is too large (eg. Cs), or if one attempts to produce a compound with stoichiometry AxC60 x > 3, (A = K, Rb or Cs), the C60 anions will no longer remain close packed, transforming to a body-centred tetragonal (bct) arrangement in the case of

A4C60, or body-centred cubic (bcc) arrangement in the case of A6C60, to relieve the congestion at the fcc tetrahedral site.

The best fits to crystallographic data for the bct M4C60 structures (M = K, Rb and Cs) 1 23 indicate that the C60 balls are merohedrally disordered, in the space group I4/mmm .

In the corresponding M6C60 structures the balls are aligned in only one of the two orientations, in the space group Im 324.

The situation in the case of Cs3C60 is unclear. Body-centred cubic structures have been 25 reported ; samples of Cs3C60 prepared in liquid ammonia and deammoniated at modest

2-6 temperatures (150°C) can be indexed as a mixture of two phases, one of which, known as the A15 phase (space group Pm 3n) is also essentially bcc with respect to the C60 26 molecules . The Cs3C60 prepared in this way is metastable and disproportionates into

Cs1C60 and Cs4C60 at higher temperatures. This accounts for photoelectron emission 27 spectroscopy (PES) results for a series of CsxC60 compounds . The A15 phase is also 28 observed in Ba3C60 prepared by vapour phase intercalation . Band structure calculations of hypothetical fcc Cs3C60 lead to the suggestion that electronic factors could perhaps play a role in the instability of this and similar compounds29. Such calculations are further discussed in § 2.2.

The relationship between the main structures described so far is best visualized in a bct representation of the unit cells, shown in Figure 2.4.

Figure 2.4: Body-centred tetragonal representation of several commonly observed fulleride structures; the C60, A2C60, A3C60 and the bottom right A6C60 structures have face-centred cubic packing of the C60 molecules. In the A6C60 structure, observed for Na6C60, the green sites are 50% occupied.

Diffraction patterns from four of the six structures illustrated in Figure 2.4 are shown in Figure 2.5, which gives a compilation of diffraction signatures of the better-known

RbxC60 phases. (We have not studied the various oligomeric forms of Rb1C60, which are summarily described in § 2.1.5).

2-7 Rb6C60

Rb4C60 Intensity

Rb3C60

C60

0.5 1.0 1.5 2.0 2.5 3.0

-1 Q = (4 π/ λ)sin θ (Å )

Figure 2.5: A compilation of diffraction patterns of the well-known RbxC60 (x = 0, 3, 4 and 6) phases. Patterns were recorded at room temperature.

2.1.3 Other Packing Schemes

A tendency towards hexagonal close packing is observed in sodium and lithium-doped fullerides, and distinct phases with hexagonal symmetry have been observed in ammoniated lithium and sodium fullerides in this work (§ 4.1). Other structures with 30 hexagonal symmetry have been observed in NaxC60(THF)y and KxC60(THF)y , as well 31 as in host-guest compounds such as I4C60 and P8C60 .

2-8 2.1.4 C60 Molecular Structure Changes

Careful Rietveld refinements allow observation of doping-induced changes in the

n- structure of the C60 molecule itself. Some distortion of the molecule from the ideal icosahedral symmetry can be anticipated in the crystal environment, though these

n- changes seem generally small. Thus changes in the structure of the C60 molecule are usually discussed in terms of the two different intramolecular C-C bond lengths. There

n- are two distinct types of C-C bond in C60 - those joining hexagonal faces of the molecule (6:6 bonds) and those joining hexagonal to pentagonal faces (6:5 bonds). Table 2.1 summarizes the available experimental refined average bond lengths of a number of different fullerides. Calculations of the bond length changes upon doping32 are also tabulated; while the values of the bond lengths are perhaps not very well reproduced, there is a clear trend for the shorter 6:6 bond to lengthen upon reduction, with the longer 6:5 bond shortening.

Table 2.1: Changes in C60 bond lengths upon doping

Compound 6:6 bond (Å) Calc.32 6:5 bond (Å) Calc. 32 33 C60 1.40(1) 1.373 1.45(1) 1.455 47 Cs1C60 1.41(1) 1.381 1.44(1) 1.452 34 [PPN]2C60 1.399(2) 1.389 1.446(2) 1.450 24 K3C60 1.400(4) 1.396 1.452(13) 1.448 35 Na2CsC60 1.43(1) 1.43(1) 36 Ba2CsC60 1.422(11) 1.414 1.440(8) 1.446 24 K6C60 1.445(3) 1.423 1.432(10) 1.445

2.1.5 Oligomeric phases

At high temperatures the salts K1C60, Rb1C60 and Cs1C60 exhibit the rock-salt structure with an fcc array of rapidly spinning C60 balls, with the cations in the octahedral sites. Several different phases may be produced from this starting structure, depending on the thermal history. Quenching of the rock-salt structure of Rb1C60 from 500 K to 273 K leads to a cubic structure which Kamarás et al. have studied by differential scanning calorimetry (DSC)37. Slow cooling (5°/min) of the 273 K structure leads to an exotherm at 235 K (interpreted as the transition of the cubic phase to an orthorhombic phase 38 consisting of C60 dimers ). Slow heating (20°/min) of the 273 K structure results in the

2-9 appearance of an endotherm peaking at 305 K (interpreted as a C60 orientational disordering transition) overlapping with a broad exotherm that peaked at 360 K (interpreted as the formation of the linear polymer phase39 depicted in Figure 2.6, in which a [2+2] cycloaddition has occurred between neighbouring balls), followed again by an endotherm (interpreted as the dissociation of the polymer into the high- temperature rock-salt phase above 380 K). A conclusion of these authors is that the dimer → polymer transition occurs via the intervention of a cubic (monomeric) phase. It has very recently been established that balls are linked in the dimeric phase by a single 40 C-C bond . The dimer/polymer phase transitions in K1C60 are also known to compete 41 with disproportionation into C60 and K3C60 .

Figure 2.6: Structure of the A1C60 (A = K, Rb or Cs) polymer. The A ions (red) are all in sites of quasi-octahedral symmetry. (This image was downloaded from a publicly accessible internet address42)

The polymeric phase may be thought of as derived from the typical fcc structure, with the polymerization occurring along one of the [110] directions (or equivalently along the [100] direction in the bct representations shown in Figure 2.4). In the polymers only the distorted octahedral sites are filled. It is curious that polymerization appears not to 43 + occur in Na1C60 ; a possible reason is that the Na ion radius is almost ideally suited to the tetrahedral site, so that the lattice contraction necessary for polymer formation cannot occur.

2-10 In the polymers, the distance between C60 centres along the direction of polymerization 39 is only 9.14 Å , compared with 10.04 Å in undoped C60. Unusually short inter-C60 distances suggestive of oligomerization (9.35 Å) have also been observed in the 44 superconductors Na2RbC60 and Na2CsC60 at pressures of ~3 kbar . Early indications that these possibly polymeric phases are superconductors45 appear to lose some of their weight in light of the recent work of Prassides et al.46. The latter workers describe the lowering of symmetry in Na2RbC60 upon cooling - from fcc (Fm3m) above ~290 K, to sc (Pa3), to orthorhombic (Immm) at temperatures between 180 and 280 K. The small lattice parameters of the non-superconducting orthorhombic phase (a = 9.3809(6) Å, b = 9.940(1) Å, c = 14.492(1) Å) are again suggestive of oligomerization. The low- temperature orthorhombic phase could not be produced in Na2CsC60 in similar experiments.

The structural and electronic changes that occur in the oligomeric phases puts a more detailed examination of these materials beyond the scope of this work. Before leaving this topic, however, we note that monomeric A1C60 samples can be synthesized by appropriate heat treatment; neutron diffraction studies show that very rapid quenching of

CsC60 from 450 K to liquid nitrogen temperatures results in a primitive cubic (Pa 3) phase analogous to that described already for the low-temperature C60 and Na2C60 structures, except that the Cs+ ions occupy octahedral rather than tetrahedral sites47.

2.2 Solid-State Band Structures

In the case of the fullerides, the frontier molecular orbitals that become the conduction electrons in the solids are the t1u highest occupied molecular orbitals (HOMOs) of the

C60 anions (§ 5.1). Bringing two C60 molecules together will result in the interaction of their respective electronic wavefunctions, such that two non-degenerate states will be formed from the two original molecular states (analogous to the formation of bonding and antibonding orbitals in the H2 molecule). The energy difference between the two new states will increase as the molecules are brought together, for overlap reasons.

The argument can be extended to a larger number of molecules, and in the case of an extremely large number of molecules, such as are found in typical crystals, there will be

2-11 a quasi-continuum of energy states (an "electronic band") within an energy range that increases as the intermolecular distances decrease.

In a crystal there will be strong orientational dependence of the electronic structure owing to the anisotropy of the interactions, and although it has not been done as part of this work, it is possible to calculate how the quasicontinuum of energy states varies throughout the Brillouin zone, whose shape depends on the crystal symmetry. Here we aim to present the results of a number of such calculations that have appeared in the literature in the context of their relevance to the electrical and superconducting properties of the fullerides.

The band structure arising from the molecular frontier orbitals in the orientationally ordered, low-temperature Pa 3 phase of C60 is shown in Figure 2.7. Such band structure diagrams show how the energy of the quasicontinuum of electronic states varies (is dispersed) in particular parts of the Brillouin zone, a given curve describing the variation in the state's energy on moving in a straight line from one symmetry point of 48 the zone to another . At general points in the Brillouin zone the t1u -derived levels (for example) are no longer degenerate, owing to the symmetry of the crystal being less than that of the molecule. Note also that the curves derived from the t1u orbitals do not overlap with those derived from the t1g or hu orbitals.

Figure 2.7: Calculated dispersion of the valence orbitals in (left) low- temperature Pa 3 C60 (a = 14.10 Å), and (right) an approximation to 49 merohedrally disordered A3C60 (a = 14.24 Å). (From Laouini et al. )

2-12 The density of electronic states at a given energy may be calculated from the dispersion of the electronic bands throughout the entirety of the Brillouin zone. In this way the dispersion calculations used to produce Figure 2.7 can be used to generate electronic density of states (DOS) diagrams such as those shown in Figure 2.8, via the relationship:

− æ∂E ö 1 ()ν = ç ÷ g è ø (2.1) ∂k ν (ie. the density of states g(ν) at energy ν is the reciprocal of the variation of the excitation energy E with respect to reduced wavevector k, measured at energy ν). A greater dispersion, such as will occur when molecules are brought closer together, is equivalent to greater variations of E within the Brillouin zone, and thus leads to lower DOS values (provided that bands do not overlap) and wider energy bands.

Satpathy et al. have calculated the electronic DOS of C60 in different molecular orientations. Their results for the t1u-derived band are reproduced in Figure 2.8, from which it is clear that molecular orientation significantly influences the shape of this conduction band. The consequences of this for superconductivity in the fullerides is discussed in § 2.3.1.

Figure 2.8: Conduction band density of electronic states for (top to bottom) merohedrally ordered C60, merohedrally disordered C60 and Pa 3 C60. Vertical lines show the energies associated with filling by one, two, three and four electrons. (From Satpathy et al.50)

2-13 By analogy to the case of the discrete molecule, the energy bands are filled up to a point at which the total electronic charge matches the total nuclear charge, and the electronic properties of the solid are governed by the electrons in the valence bands. If the resulting situation is such that bands are either completely filled or completely empty, the material will be an insulator and the minimum energy required to promote an electron from the filled band with the highest energy to the empty band with the lowest energy is called the band gap. Such a gap is said to be direct if the adjacent extremities of adjacent bands are due to dispersion at the same point in the Brillouin zone, and indirect otherwise.

In undoped C60 as well as A6C60 (A = K, Rb or Cs), the t1u-derived band is completely empty and completely filled, respectively, leading to insulating behaviour. For undoped

C60 the calculated band gap depends substantially on the procedure used, as described by Louis and Shirley51, who compare band gaps found using different calculational procedures (eg. the quasiparticle band gap is ~2.15 eV, while the local density approximation (LDA) gap is ~1.04 eV), and tabulate a number of experimental results, the average of which indicates a band gap of ~2.4 eV. Erwin and Pederson presented an 52 early calculation of the band structure of K6C60 . They calculated an indirect band gap of ~0.48 eV, the gap being ~0.54 eV at the zone centre (Γ point), though these values were anticipated to be underestimated by 20 - 50%. The valence bands were calculated to contain at most 4% K admixture, indicating almost complete charge transfer. Experiments (inverse photoemission spectroscopy, IPES)53 show a band gap of ~1 eV. 54 The same argument applies to the A15 Ba3C60 fulleride , whose t1u-derived band is completely filled55 56, leading to insulating behaviour57 with a band gap of at least ~0.35 eV58.

In the simple scenario considered so far, the addition of up to six electrons may be seen as corresponding to the filling of the valence band to an energy (the Fermi energy, εf) such that 1/6, 2/6,...6/6 of the conduction band states are filled; vertical lines in Figure 2.8 show the energies corresponding to filling by up to four electrons. Incompletely filled bands give rise to metallic behaviour, such as is seen in K3C60 and Rb3C60. The

Fermi energy will of course be associated with a density of states, denoted N(εf), as well as one or more surfaces (Fermi surfaces) within the Brillouin zone which correspond to

2-14 those parts of the dispersion surfaces with the Fermi energy. The Fermi energy density of states determines in large measure the electrical transport properties of the metal, and more importantly in the context of the present work, the superconducting transition temperature via the McMillan equation (Equation 2.5 in § 2.3.5). Experimental values 59 60 of N(εf) for K3C60 and Rb3C60 obtained in experiments and calculations vary widely -1 -1 from ~1 to ~28 states(eVC60) , though values from 10-15 states(eVC60) are typical.

One factor that potentially complicates the positioning of the Fermi level within such band structures, and therefore the calculated N(εf), is the possibility of incomplete charge transfer, associated with the presence of additional states below the Fermi level 61 formed by hybridization of alkali metal orbitals with C60 valence orbitals . It is generally accepted that charge transfer is essentially complete in the AxC60 fullerides (A = K, Rb, Cs; x ≤ 6). Although not challenging the basic consequences of this conviction, recent electron spin resonance (ESR) results indicate that the charge transfer is not quite complete, with the small fraction of electronic charge remaining on the alkali metal playing a significant role in the spin-lattice relaxation times of the 62 63 conduction electrons . Electronic structure calculations of A8(C60)3 clusters also demonstrate that the alkali metal orbitals can make a significant contribution to the valence bands (less than 10% for potassium, but greater than 25% for sodium). Further evidence for incomplete charge transfer can be found in the bcc cluster of Na atoms in 19 the octahedral site of compounds such as Na9C60 ; the edge of the cube of Na atoms refines to 3.21 Å, which is between the values for bcc sodium metal (4.24 Å) and a + hypothetical cube of contacting Na ions (2.34 Å). There is some evidence for Li-C60 9 bond formation in fcc Li2CsC60, which would be a consequence of such hybridization .

In the context of hybridization of cation states into the band structure below the Fermi level, it is appropriate to discuss briefly the situation in the alkaline earth fullerides, 18 64 which include the superconductors Ca5C60 , Ba4C60 (earlier reports indicated that the 65 66 superconducting phase was Ba6C60 ) and Sr6C60 . In these, the Fermi level lies in a band arising in part from the C60 t1g levels and in part from the alkaline earth valence 76 67 orbitals . PES results indicate that in Ba6C60, states near εf have ~25% Ba character, 58 with hybridization in both t1u- and t1g-derived bands . Such hybridization would 65 account for the apparently metallic behaviour of Ba6C60 , since complete charge

2-15 transfer could otherwise be expected to lead to complete filling of both the t1u and t1g-derived bands.

Note the assumption that the band structure does not change upon doping. It was known from an early stage that such a "rigid band" scenario was generally not appropriate for 68 the fullerides, even in the case of metallic K3C60 and Rb3C60 (a pseudo-gap appears in the conduction band density of states in these compounds, which is 0.4 - 0.5 eV wide 69 and comparable to that in the A4C60 compounds ). In fact, based on calculations such as were used to produce Figure 2.8, and under the assumption of rigid bands, we could

n- 70 expect all salts of C60 (n = 1 - 5) with the fcc structure to be metallic . It has been 71 72 known for some time that sc Na2C60 and bct K4C60 are both insulators, however, the latter having a band gap of between ~0.2 and ~0.5 eV69 72 73. The electronic properties 47 of other interesting structures, such as rapidly quenched metastable Cs1C60 (which at

1- the time of writing is the only known alkali fulleride containing discrete C60 anions),

4- 74 75 5- and recently prepared fcc structures containing C60 (NaBaCsC60) and C60 36 75 (CsBa2C60) anions remain unknown, though a band structure calculation for fcc 76 CsCa2C60 predicts it to be metallic .

Four mechanisms have been proposed to account for metal/insulator transitions in the fullerides: 1. The contribution of alkali metal states to the band structure below the Fermi level for certain metals, as already discussed, such that fullerides that might otherwise be expected to be metallic have filled bands77. 2. Jahn-Teller (JT) splitting of the molecular orbitals in the anions to such an extent that bands formed by the dispersion of these states in the solid no longer overlap. 3. Electronic dispersion in different structures could potentially lead to

splitting of the t1u-derived band into separate sub-bands. 4. Mott-Hubbard transitions. Among the fullerides, different crystal structures are often associated with metallic or insulating behaviour; whether such structural changes are a cause or an effect of the transition is of course a matter of considerable interest, but one which has not been clearly resolved at the present time.

2-16 Splitting of the t1u-derived band due to the JT effect has been put forward as the cause of 78 79 insulating behaviour in fcc Na2C60 and bct A4C60 (A = K, Rb and Cs) . While most authors have been cautious in their assignment of the causes of such behaviour78, there have been attempts to quantify it in terms of the JT effect77, as shown in Figure 2.9.

(Interestingly, an insulator-to-metal transition has been observed in Rb4C60 under pressure80.)

Figure 2.9: Electronic density of states for C60 and KnC60 (n = 1 - 4), 2- 4- showing the gap brought about in C60 and C60 salts by Jahn-Teller splitting in the anions (from Remova et al.77)

The origin of the Mott-Hubbard metal/insulator transition may be appreciated in a naive sense by considering a large number of hydrogen atoms which are slowly brought together. At large interatomic distances the electrons bind to individual nuclei, and are not free to move. If the atoms are brought closer together, and if several of the electrons are excited out of the bound states, they can (depending on the density of the free electrons) effectively screen each other from the nuclei and therefore remain free, giving rise to a metallic ground state. The effect is cumulative - once the critical electron density is reached every extra free electron assists the screening - so that the

2-17 metal/insulator transition is very sudden. The necessary critical density of free electrons may be reached by bringing the atoms sufficiently close together (since each atom potentially contributes electrons), or by thermal excitation of electrons out of bound states.

Applying this argument to the fullerides, we can anticipate that as the fulleride anions are forced further apart by the intercalation of ever larger cations, there will come a point where a Mott-Hubbard metal-to-insulator transition will occur. Probably the best evidence for such transitions in the fullerides comes from the ammoniated fulleride, + K3C60(NH3)1, which may be loosely described as a distorted fcc structure with K ions + 81 in the tetrahedral sites and [K-NH3] units in the octahedral sites . This compound, whose lattice parameters are suggestive of a superconducting transition at around 30 K (ie. at the upper end of the range of values observed for fullerides; see § 2.3.1) shows a metal-to-insulator transition when cooled below 40 K82. On the other hand, under hydrostatic pressure it can be made to superconduct84, and since superconductivity only occurs in metals, it appears that pressure-induced lattice contraction stabilizes the metallic state to lower temperatures, as expected.

The conditions associated with enhancement of the superconductivity (eg. by increasing the lattice parameter) also lead to the instability of the metallic state83, and it has been surmised that such instabilities place an upper bound on the superconducting temperatures attainable in the fullerides84.

2.3 Fulleride Superconductivity Review

Much of the interest in the fullerides has been directed towards a better understanding of 85 the superconductivity originally observed in K3C60 .

The fullerides are the first superconductors in which it has been possible to demonstrate

(by measurements of the normal state conductivity just above the Tc) that the superconductivity is three-dimensional86.

2-18 2.3.1 Relationship between structure and superconductivity

With a few exceptions, all the known fulleride superconductors are based on either the Fm 3m (fcc) or the related Pa 3 (sc) structures described in § 2.1.1. A correlation between the lattice parameter of the former structures and their superconducting temperatures was established87 soon after the initial observation of superconductivity in the fullerides. Later it was found that the Pa 3 compounds followed a different and steeper curve, whose upper experimental extremity coincides with the curve for the Fm 3m structures, as shown in Figure 2.10. The difference between these two structural types lies in the orientation of the balls, the different Tc vs lattice parameter curves for the two structures thus illustrating the sensitivity of the superconductivity phenomenon

3- to the C60 anion's orientation. It is generally believed that these structural changes exert their influence on the Tc through changes in the density of electronic states at the Fermi ε surface, N( f ), which, as described in § 2.2, is sensitive to orientational order. There ε are other theories also, however; the evaluation of N( f ) in Pa 3 Na2AC60 (A = K, Rb and Cs) by NMR studies has been used in conjunction with other data to argue that the

Tc depression in the Pa 3 materials is also related to crystal disorder and the presence of 88 paramagnetic centres in these samples . The compound Li2CsC60, with lattice parameter a = 13.998 Å89, in which the balls are considerably disordered, is not a superconductor9.

Figure 2.10: Influence of the lattice parameter upon the superconducting Tc for Pa 3 and Fm 3m fullerides of the type A3C60 (A = Na, K, Rb and/or Cs) (From Fischer et al.8)

2-19 Superconductivity has been observed in other crystal structures also, including Cs3C60 26 (essentially bcc, Tc = 40 K at 1.45 GPa) , K3C60(NH3)1 (orthorhombic distortion of fcc, 81 84 66 64 Tc = 28 K at 1.50 GPa) , Sr6C60 (bcc) Ba4C60 (bct) , Ca5C60 (simple cubic variant 18 17 of fcc) and Yb2.75C60 (orthorhombic distortion of fcc supercell) , indicating that the crystal structure is not crucial in determining whether a particular fulleride will superconduct. This strongly supports the notion that Cooper pair (§ 8.1) formation mechanism is an on-ball effect, and is effectively independent of the electronic or vibrational dispersion in the solid.

3- The Fm 3m and Pa 3 structures so far described all contain the C60 anion. The recent

n- successful synthesis of an essentially isostructural series of salts in which the C60 anion charge varies continuously for 2 < n < 3 in the Pa 3 structure, as well as n ≈ 3.3, n = 4 and n = 5 in the Fm 3m structures74 75, provides compelling evidence that at least within

3- this structural series, superconductivity is associated with the C60 anion only.

Despite this, there have been recent claims of superconductivity90 in the organic 91 ferromagnet TDAE-C60 (TDAE = tetrakis(dimethylamino)ethylene; the structure is a

1- 92 monoclinic distortion of the fcc C60 structure containing the C60 anion ) and also in 93 K1C60 . As discussed in § 2.2, the electronic bands derived from C60's t1u valence electrons in the M4C60, M5C60 and M6C60 (M = alkaline earth) superconducting phases are completely filled, the Fermi surface lying in bands formed by hybridization of C60's 67 t1g orbitals and the alkaline-earth atom states ; it is therefore perhaps inappropriate to associate fulleride superconductivity with a particular anion, other than to note that

n- occupancy of the C60 t1g orbitals corresponds to formation of C60 anions with n > 6, as might be expected from the stoichiometry of these salts if complete or almost complete metal-to-C60 charge transfer had occurred.

The variation of Tc with pressure amongst the superconducting compounds is of some interest from a mechanistic standpoint. For reasons outlined in § 2.2, one may expect the exertion of pressure on a metallic fulleride to be accompanied by a decrease in ε N( f ), which according to the McMillan equation (§ 2.3.5) would then lead to a decrease in the Tc. For compressed Fm 3m and Pa 3 fullerides this expectation is borne out by the fact that in pressure studies, the Tc vs lattice parameter falls on the respective

2-20 94 95 96 97 98 curves shown in Figure 2.10 . However, measurements of Tc vs pressure for 99 26 the superconductors Ca5C60 and YbxC60 , as well as the A15 Cs3C60 phase , show a ε small increase in Tc upon compression. A pressure-induced increase in N( f ), and thus

Tc, could possibly come about if the Fermi level occurred in the overlap region of two electronic bands; increased pressure-induced dispersion of both bands would then lead ε to an increased overlap of the bands and a consequently greater N( f ). There are other ways to account for a positive pressure coefficient of Tc, however - for example in the 26 Cs3C60 it was attributed to particle size effects and the two-phase nature of the sample .

2.3.2 Isotope effects

One of the often-used "tests" for the involvement of phonons in the superconductivity of a substance is to measure changes in the superconducting Tc upon isotopic substitution. Such an effect may be anticipated within the BCS mechanism (§ 2.3.5) because phonon energies will depend on the masses of the atoms involved. In such cases the relationship α = M Tc constant (2.2) is obeyed, where M is the isotope mass, and a value of α ≈ 0.5 may be taken as suggesting the involvement of phonons.

It is found that isotopic enrichment of natural abundance Rb (72.2% 85Rb, 27.8% 87Rb) 85 87 to almost pure Rb or Rb in Rb3C60 does not lead to a significant change in the Tc of 100 this material (α = -0.028 ± 0.036) , providing strong evidence that alkali-C60 optic phonons are not important in the fulleride superconductivity mechanism.

The fullerides K3C60 and Rb3C60 do show carbon isotope effects, however (α = 0.37 101 102 103 ± 0.05 , and 0.30 ± 0.05 for Rb3C60, and α = 0.30 ± 0.06 for K3C60 ). These values are significantly lower than 0.5, but not unusually so, and do not necessarily require the explanation of fulleride superconductivity by non-phonon mechanisms.

One other kind of isotopic substitution experiment is possible in the case of the fullerides; it is of course possible to prepare fullerides from a mixture of isotopically 12 13 102 pure C60 with isotopically pure C60. This has been done by Chen and Lieber , who 12 13 find that Rb3( C60)0.5( C60)0.5 shows a significantly greater depression of the Tc than

2-21 12 13 13 either Rb3( C0.45 C0.55)60 or even Rb3 C60. The significance of this result has been a source of debate ever since, but it is perhaps suggestive of the involvement of intermolecular phonons.

2.3.3 Superconducting Parameters of the Fullerides

There are several parameters that may be used to characterize a given superconductor, and we describe these in § 8.1. Here we give a very brief overview of the more important parameters of the fullerides.

The results given in Table 2.2 were determined by magnetic measurements of powder samples, or where possible, single crystals104 105. Muon spin resonance106 and infrared reflectivity107 have also been used in such studies.

Table 2.2: Fulleride Superconductivity Parameters

Superconductor Hc1 (mT) Hc2 (T) ξ (Å) λ (Å) 104 105 105 † K3C60 4.2 ± 0.1 17.5 45 4200 104 105 105 † Rb3C60 3.2 ± 0.1 76 20 5400 108 108 108 † RbCs2C60 6.8 ± 1.3 17 44 3300 ± 400 64 64 Ba4C60 -- 2.4 ± 0.3 116 ± 7 --

† Calculated from Hc1(0) and Hc2(0)using equation 8.2 (§ 8.1).

The much greater magnitude of λ (magnetic penetration depth) than ξ (Cooper pair coherence length) makes the fullerides extreme type-II superconductors (see § 8.1).

2.3.4 Electronic Pairing Model of Fulleride Superconductivity

Not long after the discovery of superconductivity in K3C60, Chakravarty et al. proposed a model in which the attractive interactions between electrons which result in the formation of Cooper pairs necessary for superconductivity come about as a result of on- ball electron correlation effects109.

The model used by these authors describes the tendency of electrons to form pairs in terms of an on-site Coulomb interaction parameter U, and a hopping amplitude t, which

2-22 is a measure of the ability of electrons to "hop" from one atom on the ball to a neighbouring atom. For the C60 molecule, which contains two kinds of bond (6:5 and 6:6 bonds) there are two values of t (denoted t and t', respectively, with t'/t ≈ 1.2, and t ≈ 2 - 3 eV). The ratio U/t governs the multiplet structure (see § 5) in the ground states of

n- the C60 (n = 1 - 6) anions.

n- For a C60 anion it is possible to define an electron pair-binding energy as: n =−−ΦΦ Φ E pair 2 nn-1n+1, (2.3) Φ n- n where n is the ground-state energy of the C60 anion; E pair is thus a measure of the

n anion's tendency to disproportionate. A positive E pair may be associated with a tendency

n- for disproportionation of the corresponding C60 anion. Chakravarty et al. predict that pair binding is possible on the anions for certain values of U/t, and this has been used to account for superconductivity.

Within the electron correlation theory the superconducting Tc may either increase or decrease with the lattice parameter, depending on the relative magnitudes of Epair and the intermolecular hopping amplitude, ti. For large pairing energies the Cooper pairs are already formed on individual balls, so that each ball could behave as a superconducting grain coupled to neighbouring balls through the intervening space, which behaves as a weak link (see § 8.1). In this scenario the Tc would be expected to decrease as the balls are drawn apart. On the other hand, if ti is large (and thus conducive to electronic band formation), the Tc is expected to be related to Epair and the electronic bandwidth (W) by:

∝ -(W E pair ) Tec . (2.4)

For small Epair the exponential factor becomes dominant so that Tcs will decrease with W, and thus increase as the balls are drawn apart. Equation 2.4 may be compared with the formula obtained from the electron-phonon coupling mechanism (Equation 2.11 in § 2.3.5), bearing in mind that the conduction bandwidth is inversely proportional to the

Fermi level density of states, N(εf).

One phenomenon difficult to account for by the electron correlation model is the isotope effect (see § 2.3.2). While an isotope effect is generally taken to be symptomatic of a phonon-mediated pair formation mechanism, proponents of the purely electronic

2-23 mechanisms of superconductivity have been quick to argue that significant isotope effects are not inconsistent with their models110 111 112, despite some opposition113. In an electronic correlation model an increased C isotope mass results in decreased zero-point motion, leading to smaller average intramolecular C-C distances - the consequent increase in intramolecular hopping amplitudes (t) then bring about a decrease in Epair, which reduces the superconducting Tc through Equation 2.4. In other words it can be said that the isotope effect could come about as a result of phonons modulating electronic interactions rather than being directly coupled to electrons in the sense of the phonon model (§ 2.3.5).

Symmetry-breaking Jahn-Teller distortions of the molecule, while having a deleterious effect upon Epair, are expected to be suppressed by correlation effects in this model.

2.3.5 Electron-Phonon Coupling Model of Fulleride Superconductivity

In the highly successful Bardeen-Cooper-Schreiffer (BCS)114 mechanism for superconductivity, the passage of an electron through a crystal lattice is seen as causing the polarization of the lattice along its path, the consequent ion displacements being equivalent to the excitation of a pulse of phonons. The polarization lasts long enough for another electron of opposite spin, and similar but negative wavevector to de-excite the phonons. In this way the two electrons are coupled together by their interaction with phonons to form the Cooper pairs necessary for superconductivity (see § 8.1). The first model to attempt to quantify fulleride superconductivity in terms of such a mechanism was that of Varma et al.115

Within the BCS model, the superconducting transition temperature (Tc) may be determined using the McMillan equation116:

é 1041.(+λ ) ù −ê ú D ω * T = e ë λµ−+(.)1062 λû. (2.5) c 1.2k Here D and k have their usual significance, 1 ω = åλω log ννln (2.6) λ ν

2-24 is a logarithmic average of the phonon frequencies, λν is the dimensionless electron- phonon coupling constant for mode ν,

λλ= å ν , (2.7) ν and µ* is a quantity known as the Coulomb pseudopotential, which is typically fitted to values of the order of 0.1.

th In vibrational spectroscopy the dimensionless coupling constant λν for the ν mode, of degeneracy g, frequency ω, and whose full width at half maximum is γ, may be found using Allen's formula117: 2gγ λ = , (2.8) ν πω2 () ε N f though there has been some question concerning the applicability of the formula to the fullerides, since it generally only applies when the phonon energies are much smaller than the electronic bandwidth (ω << W)123. Gas phase photoemission spectra from the

1- 118 C60 anion have also been used to evaluate electron-phonon coupling constants . In addition, there have been a number of calculations of these constants115 119 120 121 122.

Comparison between different models may be made using the coupling constant = λε() VNννf , (2.9) as shown in Figure 2.11. While the Raman data favour the low energy modes, most theoretical models tend to place emphasis on the high energy modes. As discussed in § 6.2, the inelastic neutron scattering results recorded as part of the present work are not readily analysed in terms of the Allen formula, but do show evidence for the involvement of the Hg(2) mode, and perhaps also the Hg(1) mode.

2-25 0.05

PES 0.04 Raman Asai Schluter (eV) (eV) Antropov 0.03 Varma Faulhaber

0.02 Coupling constant V Coupling constant 0.01

0 270 432 709 773 1100 1248 1425 1572 Hg(1) Hg(2) Hg(3) Hg(4) Hg(5) Hg(6) Hg(7) Hg(8)

Figure 2.11: Electron-phonon coupling constants for the eight Hg modes. Photoemission (PES)118 and Raman123 results are experimental; the others were calculated115 119 120 121 122.

Equation 2.9 and the expression

V=å Vν (2.10) ν allows the following crude but useful87 approximation of Equation 2.5:

æ 1 ö −ç ÷ è ε ø ∝ ω VN( f ) Tcloge . (2.11)

It is not unreasonable to suppose that the phonons that scatter the conduction electrons and bring about electrical resistivity are also the ones that can couple to them in a BCS- type superconductivity mechanism. The temperature dependence of the normal state resistivity of the fullerides is therefore interesting in that it can give some idea of the phonon frequencies that are involved in electron-phonon interactions. Figure 2.12 shows the temperature dependence of the normal state resistivity of single-crystal K3C60 as measured by Crespi et al.124. In addition to the intramolecular modes usually implicated in superconductivity mechanisms, the resistivity results show evidence for

2-26 coupling to low energy phonons also; the best fit to the data shown includes coupling to a phonon of ~100 cm-1 (though the data cannot be used to give an accurate energy).

Figure 2.12: Temperature dependence of the normal state resistivity of 125 single crystal K3C60 ; the dotted curve is a fit using the electron-phonon coupling constants of Jishi and Dresselhaus126, the dashed curve uses the results of Schlüter et al.120, and the solid curve uses the results of Varma et al.115 with added coupling to a ~100 cm-1 phonon. (From Crespi et al.124)

2-27 2.4 References

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2-32 3. Synthesis and Superconductivity

3.1 Preparative Methods

Several techniques have been applied to the synthesis of fullerides, both in the present work and in the general literature. For the purposes of the neutron scattering described in § 6, the conventional technique involving direct reaction of C60 with alkali metal vapour was the method of choice. Variations on this theme have been employed in the literature, which we briefly discuss. In what follows we focus on the use of liquid

n- ammonia as a solvent, such syntheses having led to the observation of the C60 (n = 2 - 6) anions' absorption spectra (§ 5) as well as several new crystal structures (§ 4). Other solution-phase syntheses, including electrocrystallization, are summarily reviewed.

3.1.1 Vapour-phase intercalation

The conventional method of preparing binary and ternary alkali fullerides has been to mix stoichiometric quantities of C60 with the alkali metal(s), and heat the mixtures in evacuated pyrex or quartz tubes for about a week to temperatures of 300-400°C. The technique generally produces satisfactory results and is widely used in the literature and throughout this work. It may be used for the less volatile alkaline earth metals also1 2 3 4, 5 and has been used in the preparation of Yb2.75C60 , although the higher temperatures required to promote reaction become critical as decomposition of the C60 is liable to occur at temperatures much in excess of 700°C.

Despite the general applicability and widespread use of this synthesis, difficulties in the preparation of large samples of Na2C60 and Rb4C60 for TFXA measurements (see § 6),

3-1 as well as in the preparation of a sample of Rb3C60 used as a superconductivity standard (§ 3.2), may reflect problems with the technique, which we describe here.

The preparation of the TFXA Rb4C60 sample began with the addition of 28.70 cm of 2 0.02219 cm bore area Rb-containing capillary to 2.0512 g of degassed C60 in a pyrex tube under high purity argon, thus anticipating a stoichiometry of Rb4.01C60. This was then evacuated to ~10-4 τ, sealed with a torch, and placed in a furnace at 340°C for five days, with daily thorough mixing of the sample to ensure homogeneity. After this time, the product mass (2.979 g) indicated an overall stoichiometry of Rb3.81C60. Fits to synchrotron diffraction data of this material showed it to contain substantial amounts of

Rb6C60 and smaller amounts of Rb3C60, as well as the desired Rb4C60, and indicated an overall stoichiometry of Rb4.77C60. On the assumption that the intercalation process had led to the decomposition of part of the C60 to an amorphous, Rb-free component, a further 0.319 g of C60 was added to return the composition of the crystalline fraction from Rb4.77C60 to Rb4.00C60. The sample was then annealed at 330°C for a further six days. Refinement of the final product is described in § 4.3, and indicates an overall crystalline phase stoichiometry of around Rb3.67C60. This composition of stoichiometric crystalline phases, and the quantities of reagents used in the preparation, suggest that the sample contains 30.1 wt.% Rb3C60, 66.4 wt.% Rb4C60, and 3.5 wt.% of the supposed amorphous Rb-free component. The sample shows a magnetic susceptibility transition at ~30 K, as expected for superconducting Rb3C60 (see § 3.2).

Refinements of the superconducting Rb3C60 standard, also prepared by vapour-phase intercalation, again indicated an unexpectedly high rubidium content: Rb4.22C60. These 6 results seem curious in view of the Rb3C60 synthetic study of Schlueter et al. , who report the presence of only Rb1C60 and Rb3C60 in synthetic conditions closely matching our own (ie. they observe overall stoichiometry less than anticipated, as could possibly occur if the alkali metal were partly oxidized prior to intercalation).

The TFXA Na2C60 sample was similarly prepared by heating degassed C60 and sodium metal together at 340ºC for 5 days with vigorous daily agitation to ensure homogeneity. Again, the product is known not to be phase pure, but in this instance no attempt was made to modify the stoichiometry.

3-2 The vapour phase method has been applied to the synthesis of millimetre-sized single crystals of the fullerides7. The intercalation process leads to some deterioration of the 8 9 original crystal, however, particularly in the case of Rb3C60 .

A closely related technique that may become important in the preparation of fullerides 10 11 involves the use of plasmas induced in inert gases . The reaction of C60 with alkali metal vapour is essentially the same, the principal advantage of the plasma technique being in the speed of preparation, the process requiring only a few seconds.

3.1.2 Preparation via decomposition of alkali-containing compounds

A modification of the vapour-phase intercalation involves the use of compounds that decompose upon heating to yield the alkali metal and an inert substance which can be readily removed from the product. Such compounds include hydrides of the alkali 12 metals (eg. NaH ), borohydrides (eg. NaBH4), alloys with other metals - particularly mercury, which can be readily distilled away from the solid (eg. KHg and RbHg13 14, and 12 15 16 Na5Hg ), and azides (eg. NaN3 and CaN6 ). In terms of preparation these substances offer several advantages over intercalation using the pure metal - all are relatively hard solids and may be manipulated with comparative ease; they are less reactive than the pure metal; and the higher molecular weight of the compound reduces errors in weighing. The alkali azides in particular are not air-sensitive, and may easily be purified by recrystallization from water or ethanol.

One complicating aspect of the use of azides in syntheses is the formation of MαNβ 17 18 clusters in the resulting solid . Fullerides prepared from NaN3 are particularly interesting as, depending on the conditions of preparation19 20, they may superconduct, in contrast with the insulating Na3C60 prepared by other syntheses. Structural characterization of these structures appears to be still in progress at the time of writing21. Magnetic properties of the superconductor have also been investigated22. Superconductivity has also recently been observed at 15 K in an fcc phase 23 (a = 14.365 Å) formed by intercalation of NaH into C60.

3-3 3.1.3 Preparation by titration

Stoichiometry control was a problem in the synthesis of fullerides at an early stage in the experimental work, since C60 was not commercially available at the time and could only be produced in small quantities. However, it was known that the fulleride anions in solution had characteristic absorption spectra in the near infrared (NIR)24, and consequently we tried to exert some control of stoichiometry by in situ spectrophotometric titration of solutions containing fullerides, which could then be evaporated to dryness, and the solvent-free fulleride recovered by heating under vacuum and subsequent annealing.

The choice of solvent was governed by a number of considerations - most importantly it would have to be stable to the very strong reduction potentials necessary for the production of fulleride anions; other highly desirable properties would be a simple molecular structure (the solvates themselves might well turn out to be of interest - as has indeed been the case), high volatility (to help facilitate its removal from solvates if necessary), the ability to act as a solvent for at least one of the reactants as well as the products, and high purity.

Preliminary experiments indicated that liquid ammonia was a very good solvent for

Rb3C60, and since it satisfies the other requirements outlined above, it was the solvent of choice. In particular its ability to dissolve the alkali metals without appreciable decomposition was useful. At the same time as the experiments described here25, liquid ammonia was being used elsewhere in the successful synthesis of a variety of alkali metal fullerides26.

The considerable technical difficulties in performing spectrophotometric titrations in liquid ammonia were overcome using the apparatus shown in Figure 3.1. In essence, two 50 mL burettes were attached to a 100 mL reaction vessel, and a 1 mm quartz cell was appended to the latter for spectrophotometric measurements. The burette and reaction vessel were jacketed to allow circulation of cold methanol at -60ºC. All taps coming in contact with the metal-ammonia or fulleride-ammonia solutions were glass shafted with silicone O-rings; teflon reacts with the metal-ammonia solutions, and the

3-4 Figure 3.1: Liquid ammonia titrator, side and front views.

3-5 usual neoprene O-rings become too brittle at the low temperature of operation. A support frame was also constructed to permit raising and lowering of the apparatus into the beam of a Cary 5 spectrophotometer.

Prior to each titration, the apparatus was allowed to stand overnight in aqua regia, before successively rinsing with water, ethanol and ether. A small quantity (1 - 2 mL) of trimethylsilylchloride was then admitted and the apparatus allowed to stand for a few hours, in order to deactivate the hydroxyl groups at the surface of the glass and thus minimize unwanted reactions that might otherwise occur during the titration. After this treatment, the titrator was rinsed twice with dry ether and evacuated before admission to a high-purity (< 0.5 ppm O2) argon drybox, where a known quantity of alkali metal was placed in one of the burettes. A double-glazed nitrogen-cooled cell block was built from polystyrene foam to minimise stray light and control bumping of the cuvette contents, a process which allowed mixing with the contents of the reaction chamber between measurements.

To help ensure a homogeneous suspension of C60 in the liquid ammonia, the C60 was dissolved in toluene and the solution poured into a similar volume of briskly stirred, ultrasonicated ether. After several minutes the mother-liquor could be decanted off the precipitated C60, which proved to be sufficiently finely divided for the purposes of the titration. Pumping on the product whilst warming to about 100ºC for half an hour 27 helped ensure that no ether had intercalated into the C60 . The suspension of C60 and the dissolution of the alkali metal were facilitated by the constant gentle boiling of ammonia that occurred at the titration taps.

Table 3.1 summarizes the nine titrations performed in the apparatus. It can be seen that the agreement between the anion observed in the final spectrum and the final titre is quite poor. The possible reasons for this will be discussed shortly, but for the moment it suffices to say that the final spectrum is felt to be a better indication of the alkali:C60 ratio than the titre. A large number of Rb2C60 titrations were performed, owing to the false observation of superconductivity at 56 K in one of these samples (see § 3.2).

3-6 Table 3.1: Summary of the liquid ammonia titrations

Alkali Final Final Titre Alkali Final Final Titre Metal Spectrum (A:C60) Metal Spectrum (A:C60) 3- 3- Li C60 3.7 Rb (#4) C60 (problems) 2- 4- Na C60 5.1 Rb (#5) C60 5.5 2- 4- Rb (#1) C60 2.7 Rb (#6) C60 5.9 2- 5- Rb (#2) C60 3.0 Rb (#7) C60 8.3 2- Rb (#3) C60 3.0

The observation of isosbestic points in the titrations indicates an equilibrium between two and only two absorbing species; such points are characteristic of the species involved. The relevant values are tabulated in Table 3.2, after allowing for the changing concentration of C60 in the solutions. Too few spectra were obtained to be able to

5- 6- confidently assign isosbestic points for the C60/C60 couple.

Table 3.2: Isosbestic points for C60 anions in liquid ammonia

Couple Isosbestic Points (cm-1) 2- 3- C60/C60 10340, 12560, 28090 3- 4- C60/C60 8120, 9840, 13530, 26610, 29470 4- 5- C60/C60 9340, 13550, 25400

1- While attempts to generate the C60 anion in liquid ammonia with lithium, sodium and rubidium as reductant all failed, with sodium and lithium it was possible to

6- spectroscopically observe all the higher anions up to and including C60 . Attempts at further reduction resulted in the appearance of the solvated electron spectrum. The spectra were independent of the alkali metal used. Poor solubility made measurement of the higher anion spectra difficult when using rubidium as the reductant. Particularly good spectra were obtained from the sodium titration, and these are presented and discussed in § 5 (see Figure 5.2) where we discuss the electronic structure of the anions.

The relationship between the titre and the observed average anion charge (estimated from the spectra) for the sodium titration is shown in Figure 3.2. Data were collected over a period of 8 hours, the sodium having been dissolved in liquid ammonia for an additional 5 hours. There is clearly not a one-to-one relationship between the titre and observed average anion charge (probably due to settling of the C60 in the burette), though the majority of points fall in a linear region whose slope is only slightly less than

3-7 would be expected for a one-to-one correspondence. Possible explanations include partial oxidation of the alkali metal solutions, perhaps owing to gradual leakage of air over the very considerable periods of time involved (typically around 12 hours), or slow formation of alkali amides. The first possibility is unlikely given the linearity of the plot and the comparatively low scatter of points; the second option seems equally improbable given that ammonia solutions of alkali metals may be stored for several days with only a few percent decomposition28 and that x-ray diffraction patterns of the products did not suggest the presence of amides. A more likely explanation is that settling of the C60 suspensions led to the "concentration" of C60 near the tap in the burette being higher than it would have been in a homogeneous suspension, leading to a smaller than expected C60 titre for the production of a particular anion. The only points that do fall approximately on the ideal line are those in which the spectrum consisted

2- only of the C60 anion, but the anomalous positions of these points with respect to the

1- others suggest that the C60 anion probably disproportionates in liquid ammonia.

6

4

2 Estimated Average Anion Charge Anion Average Estimated

0 0 2 4 6 8 10 12

Na:C 60 from Titre

Figure 3.2: Estimated observed average anion charge versus Na:C60 ratio from titres. The line corresponding to the expected 1:1 relationship between the two is marked, while the illustrated line of best fit has y = 0.778x - 2.08. The data point at the extreme upper right corresponds to C60 in equilibrium with solvated electrons; this and the points whose Na:C60 ratio was less than 4 were not used in the fit.

3-8 1- The failure to observe the C60 spectrum is surprising. A possible explanation is the

1- formation of insoluble polymeric salts with the alkali metals, but the C60 anion is clearly 29 observed in a very wide range of other solvents (see also § 5) - furthermore Na1C60

30 1- appears not to polymerise . Zhou et al. have demonstrated electrochemically that C60 is soluble in liquid ammonia solutions of LiI and KI31, although the conclusions drawn from the use of LiI electrolyte in the latter report seem to be at odds with the present

2- 3- 4- 5- 6- titration results. We find that spectra of C60 , C60 , C60 , C60 and C60 are clearly observable when using lithium as the reductant with comparable C60 concentrations -3 (typically ~10 mol/L). Zhou et al. report that electrochemical reduction in LiI/NH3

5- solutions proceeded only as far as C60 before the appearance of the solvated electron

1- 2- spectrum, and that only the C60 and C60 anions were particularly soluble. Other than invoking the involvement of the iodide anion and/or unknown electrogenerated species in the latter report, a reconciliation of our results with theirs seems unlikely.

Titration products were recovered by laying the titrator on its back, draining out the coolant fluid, and vacuum distilling the ammonia back into a receiver flask on a vacuum line via a piece of flexible tubing. In this way the solid product was deposited at the back of the reaction vessel (see Figure 3.1). After evaporation of all the ammonia, the coolant jacket was rinsed with acetone and dried by a stream of air, before admitting the titrator to a high-purity argon drybox (< 0.5 ppm O2). Removal of the tap labelled "Reaction vessel entry/pressure release tap" in the illustration allowed the product to be carefully scraped off the rear wall of the reaction vessel with a long spatula. Products thus recovered were stored under argon in stoppered glass tubes.

X-ray diffraction patterns of the products were generally poorly crystalline, and were compounded by instrumental difficulties; Figure 3.3 shows one of the better results. Although many such patterns were recorded, few were of sufficient quality compared to the patterns obtained from the "bomb" products (see § 3.1.4) to be of any use for structure analysis. The only exceptions to this were the diffraction patterns obtained from the product of the lithium titration, which are described in § 4.1 and published by us in reference [32].

3-9 Intensity

0 10 20 30 40 50 60 70 2θ (degrees) Figure 3.3: Diffraction pattern (λ = 1.5418 Å) of a titration product (Rb~4.9C60(NH3)x). This pattern is one of the better ones obtained by the titration synthesis.

3.1.4 Bomb synthesis

As a synthetic approach the titration method had a number of drawbacks, principal among them being the time involved in setting up and performing an experiment and the small amount of generally poorly crystalline material that could be obtained from a particular run (typically ~30 mg). On the other hand, it appeared that liquid ammonia had a lot to offer as a synthetic medium. This, and the increasing availability of

C60 from commercial sources, prompted efforts to develop a simpler method of preparation in this solvent.

Figure 3.4: The "bomb" apparatus used for the liquid ammonia synthesis of several fullerides.

3-10 Later syntheses were carried out in the apparatus shown in Figure 3.4. The feature shown as constriction #1 allows the "U"-shaped section to be sealed with a flame after the addition of appropriate quantities of C60 and alkali metal and a few millilitres of sodium-dried liquid ammonia. Although the resulting "U"-sections were generally capable of withstanding the internal pressures developed at room temperature, being constructed of heavy pyrex tubing (2 mm thick glass, 8 mm bore), three such "bombs" exploded - without incident as appropriate precautions had been taken - at the sealed constriction after being maintained at room temperature for several hours. Temperatures in subsequent experiments were not allowed to rise much above 0°C by use of an ice-water slurry. In use, the apparatus is small, fully sealed and under considerable positive internal pressure, guaranteeing low contamination from both internal and external sources. In some instances, distillation of ammonia within the sealed bombs during preparation allowed sufficiently dilute solutions to be made for NIR spectra to be measured, which serves to identify the species present.

The design of the apparatus allows a crude form of extraction as follows. The entire contents of the "bomb" were tipped into arm "A" and allowed to settle. The supernatant liquor could then be carefully decanted into arm "B", after which the ammonia was distilled back into arm "A" by establishing a slight temperature gradient between the two. This cycle was typically repeated 7 or 8 times, until the freshly distilled ammonia ceased to dissolve significant amounts of material. At this point a final distillation was performed, leaving the liquid ammonia and any insoluble material in one arm, and the ammonia-soluble fulleride in the other. The arm containing the liquid ammonia was then cooled in a CO2-ethanol slurry, and the arms separated by sealing constriction #2. Further manipulation of the arm containing the product was performed in a high-purity argon drybox (< 0.5 ppm O2), while the arm containing the solvent ammonia was disposed of safely.

The products synthesized in this way are summarized along with their structural characterization by X-ray diffraction and other techniques in Table 4.1 of § 4.1. Other authors have prepared fullerides in liquid ammonia using similar techniques26. The 33 preparation of the A15 phase of Cs3C60 as well as the very recent preparation of 34 Ba2CsC60 testify to the versatility of our method.

3-11 Ammonia desorption measurements were carried out by heating a known mass of the product to 300°C under static vacuum and collecting the evolved ammonia in a liquid nitrogen trap. The amount of ammonia was then determined volumetrically by noting the pressure exerted when the known trap volume was warmed to ambient temperatures. Products obtained by this process (referred to as deammoniation elsewhere in this work) were annealed at 340ºC for periods of up to a week prior to examination by X-ray diffraction (see § 4.2).

3.1.5 Preparation in Organic Solvents

There have been attempts to synthesize fullerides in solvents other than ammonia. In this context, an extremely useful compilation of C60 solubilities in a large range of liquids has been made by Ruoff et al.35. Attempts to use toluene36 37 in the synthesis of

Rb3C60 resulted in low superconducting fractions, presumably owing to the poor solubility of C60 anions and formation of Rb1C60 in this non-polar solvent.

Bezmelnitsyn et al. have studied the rate of reaction of C60 and C70 with sodium metal in toluene at room temperature38. The use of a more polar benzonitrile-toluene solvent system has been described by Schlueter et al.39. Intercalation of solvent in the products of such syntheses can also occur, for example in AxC60(THF)y (A = alkali 40 41 42 43 metal) , [Na(crown)(THF)2]xC60 (x = 2, 3; crown = dibenzo-18-crown-6), 44 45 [BTTPI]3C60Cl(CH3CN), [BTTPI]3C60(CH3CN)2 , BTTPI = (bistriphenylphos- 46 phine)imine [Cr(TPP)]C60(THF)3 (TPP = tetraphenylporphinate) and 47 [CoCp2]C60(PhCN) (Cp = cyclopentadiene) .

3.1.6 Electrochemical Intercalation and Electrocrystallization

Experiments in which lithium has been intercalated into C60 in a solid state electrochemical cell using a lithium perchlorate - polyethyleneoxide electrolyte film have demonstrated that lithium can be intercalated into C60 in this way, with possible 48 formation of phases corresponding to LixC60 (x = 0.5, 2, 3, 4 and 12) . Electron diffraction has since been used in attempts to characterize some of the phases formed49.

3-12 Electrocrystallization also offers a potential route to the preparation of fullerides, as 50 51 shown by the preparation of single crystals of [N(P(C6H5)3)2]C60 , [Ru(2,2'- 52 53 54 bipyridine)3](C60)2 , Ph4PC60.Ph4Cl and (Na,K)xC60(THF)y .

3.2 Superconductivity Measurements

The primary goal of the superconductivity measurements in this project was to determine whether or not samples were superconductors, and if so, to establish their superconducting Tc. Reliable measurement of such transitions turned out to be considerably more difficult than originally anticipated, the most trustworthy results consequently being obtained very late in the study. Initial attempts to use low-field microwave absorption (LFMA) met with some success, but later led to reproducible observations of what appeared to be superconducting transitions in samples that were highly unlikely to be superconductors. This necessitated the development of an alternative and relatively simple, but reliable technique, utilizing the sudden magnetic susceptibility changes associated with superconducting transitions.

3.2.1 Low-Field Microwave Absorption

The LFMA technique for measuring superconductivity has been widely used, but is not well understood. Some workers have attributed the absorption of microwave energy to the viscosity of fluxoids that may occur in type-II superconductors55, others to the damped motion of magnetic flux at intergranular weak links56, and the AC Josephson effect may also contribute to the absorption (see § 8.1; the weak links are the boundaries between superconducting crystallites). Regardless of the mechanism(s) responsible for the absorption, the technique is sensitive to very small quantities of superconducting 57 material and may be used to distinguish multiple phases with different Tcs . It was in 58 fact used in the discovery of superconductivity in K3C60 .

In this technique, a klystron is used to supply microwave power to a resonating cavity containing the sample and a dummy load as part of a microwave "bridge". Microwaves reflected from the cavity and dummy load can be attenuated, phase shifted and recombined so as to be of equal magnitude and opposite phase when they impinge upon

3-13 a crystal diode, so that a null rectified signal results at bridge balance. In such a bridge circuit, the diode is a sensitive detector of changes in the amplitude and phase of microwaves reflected from the sample cavity, such as will occur when the sample absorbs energy having become superconducting. The absorption of microwave energy is magnetic field dependent, so that modulation of the magnetic field (typically at 100 kHz) produces a signal at the modulation frequency which may readily be amplified using a lock-in amplifier.

Figure 3.5: The design of the probes used in the measurement of superconductivity by microwave absorption. The expansion shows the modifications that were made for measurements of changes in the radio frequency susceptibility.

3-14 Measurements were performed at 9 GHz (X-band) using a Varian V4502 microwave electron spin resonance spectrometer. The sample was mounted in an Oxford Instruments flow cryostat cooled by a stream of liquid helium boil-off gas, the temperature of which was controlled by an Oxford ITC4 temperature controller. Sample temperatures were monitored using an indium-soldered Au 0.7% Fe/chromel-P thermocouple built into the Suprasil sample probe shown in Figure 3.5. A small quantity of helium gas (typically ~10 τ) added to the probe helped ensure proper thermal equilibration of the sample and thermocouple.

The detected signal and thermocouple voltages were recorded as the sample was warmed or cooled through the superconducting transition using a data-logging system. Thermocouple voltages were subsequently converted into temperatures using polynomial fits to the calibration curves for the particular thermocouple type. Temperatures measured in this way are believed to be accurate to within ±1 K.

55 While measurements by us on Rb3C60 using this technique gave a Tc of ~30 K, the results on other compounds were not readily interpretable. For example, in Figure 3.6 and Figure 3.7 we show the LFMA trace of one of the Rb:C60 titration products before and after removal of intercalated NH3 by heating and pumping. Both traces were reproducible with the same sample in the same probe, and were independent of whether the sample was being warmed or cooled during data collection. These results would lead us to suppose that prior to removal of intercalated ammonia, the sample (whose

2- final NIR spectrum was of C60, the final titre indicating Rb:C60 = 3.0:1; see § 3.1.3) became superconducting at ~13 K, which was raised to ~27 K by the formation of the well-known Rb3C60 phase after deammoniation (verified by X-ray diffraction).

Several samples, however, showed what appeared to be clear superconducting transitions at unexpected temperatures, leading us to doubt many of the observed "superconducting" transitions. The most striking example is shown in Figure 3.8, which is the LFMA trace of an extremely small quantity of Rb2C60 (2-3 mg) prepared in liquid ammonia, and which had been accidentally briefly exposed to air. The very sharp and strong transition at 56 K was readily reproduced on different occasions within the same probe and with the same sample, and generated considerable excitement as well as

3-15 several failed attempts to reproduce the sample - but was later also observed when the sample was tipped into the side arm of the measurement probe! Signal (arbitrary units)

010203040 Temperature (K) Figure 3.6: Temperature dependence of the microwave signal from one of the Rb:C60 titration products (see text). Signal (arbitrary units)

010203040 Temperature (K) Figure 3.7: Same sample as Figure 3.6, after heating at 300ºC for 1 hour under static vacuum (liquid nitrogen trap). The diffraction signature of the material was that of Rb3C60.

3-16 Other samples had also shown evidence for Tcs well above the highest reported at the 59 time for fullerides in the literature (33 K for RbCs2C60 ), so there was good reason to suspect the experimental technique. Thermocouple calibration, which led to difficulties in later experiments (see § 3.2.2), was clearly not the problem, there being reasonable agreement between the lowest temperatures observed and the temperature of boiling helium (4.2 K), as well as between the expected and observed temperature of liquid nitrogen (77 K). Signal (arbitrary units)

0 1020304050607080 Temperature (K) Figure 3.8: Temperature dependent microwave absorption of an air-exposed sample of Rb2C60 prepared in liquid ammonia; the observed transition turned out to be a property of the probe, not the sample (see text).

Several attempts to determine the cause of the unexpected transitions ultimately failed. For example, a particular empty probe was found to show a transition at around 47 K. With the probe removed from the microwave cavity sufficiently so as to be largely out of the microwave field, but still able to measure the temperature, the transition was almost completely absent. A short length of chromel-P thermocouple wire blackwaxed to the bottom of the probe so as to be in the microwave field while the thermocouple itself was largely out of the field showed no detectable transition. These results demonstrated that the transition was in some way related to the probe and that neither the cavity nor the chromel-P thermocouple wire was likely to be involved in the observed transition. The gold wire is also unlikely to be involved, leaving only the Suprasil glass tube, the Eccobond 286® epoxy cement and the indium solder as possible

3-17 suspects. Other probes, in which the indium solder was eliminated by spot-welding the thermocouple, also showed unexpected “superconducting” transitions. There is thus an unexplained loss mechanism in the Suprasil and/or epoxy cement, but further pursuit was deemed fruitless. For these reasons we are inclined to distrust the results obtained using LFMA.

3.2.2 Radio Frequency Susceptibility Measurements

An independent check on the observed transition temperatures was clearly desirable. An alternative approach to the LFMA measurements was to perform susceptibility measurements at radio frequencies (RF), which although less sensitive and so requiring considerably larger samples (typically ~200 mm3), was insensitive to the relatively weak transitions due to the materials used in the construction of the probe. The RF measurements had the added advantage of requiring only common electronic components, enabling rapid modification where necessary.

The measurements that were performed detected a change in the magnetic susceptibility at the onset of superconductivity. A solenoid buried in a medium with magnetic susceptibility κ, has an inductance given approximately by: Nr22 L ≈+µκ()1 (3.1) 0 9r + 10l where the solenoid has length l with N turns of radius r. When in parallel with a capacitor of capacitance C, the resulting inductor-capacitor (LC) circuit is well known to resonate at the frequency ω, related to L and C by 1 ω = (3.2) LC It follows that the resonant frequency of an LC circuit will change if there is a change in the magnetic susceptibility of the core of the inductor, such as occurs at a superconducting transition. In the case of a superconductor, we expect flux exclusion due to the Meissner effect (see § 8.1), which leads to κ < 0 below the Tc; this will translate to a sudden increase in the LC circuit's resonant frequency.

3-18 The inset of Figure 3.5 shows how the probe was modified for the RF measurements. The most significant features of the design are the positioning of the thermocouple, which enters above the coil to minimise coil-to-thermocouple capacitance and consequent loss of RF energy to the thermocouple circuit, and the use of very thin (0.05 mm) brass shim to help thermally isolate the coil. Loss of RF energy through the thermocouple is further minimised by insertion of an adapter plug containing a pair of inductors (one for each thermocouple wire) into the thermocouple connection socket shown in the diagram. The thermocouple is spot-welded at its tip, covered with a thin protective layer of Eccobond 286® epoxy resin, and brought to the centre of the coil. Samples were loaded into the probe under high-purity argon, which was then replaced by helium to ensure good thermal contact between the thermocouple and sample.

Cryogenic temperatures were obtained by passing the boil-off gas from liquid helium through a resistive heater in a flow tube. Temperature control was achieved by balancing of the flow rate and the power dissipated in the heater. Even for very low flow rates, temperature equilibration times were short (typically only a second or so), and by manual adjustment it was easily possible to control the sample temperature to within a degree of a desired value. (A feedback loop was initially employed to keep the temperature constant, but problems matching the loop's response time with the sample equilibration time led to dramatic temperature oscillations, so it was abandoned.)

A Robinson oscillator60 was used to excite oscillations in the LC circuit. Depending on the sample, the oscillator could be made to function within a range of frequencies (typically ~5 - 20 MHz), as observed on a frequency counter and measured on an oscilloscope. In all samples except the lead shot used as a thermocouple calibrant (see below) stable oscillation could be obtained with a completely filled coil by adjusting the capacitance in the LC circuit. Time did not permit the setting up of a data-logging system, so measurements of thermocouple voltage and oscillator frequency were entirely manual.

A sample of Rb3C60 was prepared for use as a superconducting reference (see § 3.1.1). Figure 3.9 plots a large number of frequency vs temperature measurements for this sample, which show a very clear transition at 33 K, a few degrees higher than expected 6 for Rb3C60 (Tc = 28 - 30.5 K ). That this was due to faulty thermocouple calibration

3-19 was confirmed by noting the lowest achievable temperature (8.9 K; helium boils at 61 4.2 K), and the observation of superconductivity in lead shot at 10.1 K (Tc = 7.2 K , see Figure 3.10). The relatively small frequency changes in Figure 3.10 are due to the small amount of lead shot necessary to enable oscillation. Figure 3.9, Figure 3.10 and Figure 3.11 show the measured temperatures without the ~3º subtraction.

14.6

14.4

14.2

Resonance Frequency (MHz) Resonance 14.0

0 10 20 30 40 50 60 Temperature (K) Figure 3.9: Resonant frequency vs temperature for an LC circuit containing Rb3C60 within the coil. The difference between the observed (33 K) and expected (~29 K) Tc is due to thermocouple problems (see text).

16.12

16.10

16.08

16.06

16.04

16.02

16.00

9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 Temperature (K)

Figure 3.10: Superconductivity in a lead sample. The observed Tc (10.1 K) is greater than the known value (7.2 K), indicating that the thermocouple reads a somewhat higher value than its calibration curve would suggest.

Interestingly, even the empty LC probe also showed evidence for a susceptibility transition between 50 and 60 K (see Figure 3.11), though this was negligibly weak

3-20 compared to the superconducting transitions already described. The overall gradual shifting of the resonance frequency in the figure may be attributed to changes of the resistivity and geometry of the coil and probe with temperature.

16.62

16.60

16.58

16.56

16.54

16.52 Resonance Frequency (MHz) Resonance

16.50 0 20 40 60 80 100 Temperature (K) Figure 3.11: Resonant frequency versus temperature for the empty probe, showing a slight susceptibility anomaly between 50 and 60 K.

Susceptibility measurements with the RF technique were made on samples of

Na3C60(NH3)6 (a sample containing 17.0 wt.% Na2C60(NH3)8 impurity), Rb2C60(NH3)x and Rb4C60(NH3)2 (diffraction patterns shown in Figure 4.10), the TFXA Na2C60, and the TFXA Rb4C60. Of these, only the TFXA Rb4C60 sample showed a clear susceptibility transition, which was at the same temperature as the Rb3C60 standard shown in Figure 3.9 and confirms the presence of Rb3C60 contamination (see § 3.1.1). The other samples showed only the susceptibility anomaly in Figure 3.11 at temperatures above 7 K (after taking into account a conservative 2º for the thermocouple calibration described earlier.).

3-21 3.3 References

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3-22 30 T. Yildirim, J.E. Fischer, A.B. Harris, P.W. Stephens, D. Liu, L. Brard, R.M. Strongin and A.B. Smith III, Phys. Rev. B, 71, 1383 (1993) 31 F. Zhou, C. Jehoulet and A.J. Bard, J. Am. Chem. Soc., 114, 11004 (1992) 32 R. Durand, W.K. Fullagar, G. Lindsell, P.A. Reynolds and J.W. White, Mol. Phys., 86, 1 (1995) 33 T.T.M. Palstra, O. Zhou, Y. Iwasa, P.E. Sulewski, R.M. Fleming and B.R. Zegarski, Solid St. Commun., 93, 327 (1995) 34 A.C. Duggan, J.M. Fox, P.F. Henry, S.J. Heyes, D.E. Laurie and M.J. Rosseinsky, Chem. Commun., 1191 (1996) 35 R.S. Ruoff, D.S. Tse, R. Malhotra and D.C. Lorents, J. Phys. Chem., 97, 3379 (1993) 36 H.H. Wang, A.M. Kini, B.M. Savall, K.D. Carlson, J.M. Williams, K.R. Lykke, P. Wurz, D.H. Parker, M.J. Pellin, D.M. Gruen, U. Welp, W.-K. Kwok, S. Fleshler and G.W. Crabtree, Inorg. Chem., 30, 2838 (1991) 37 H.H. Wang, A.M. Kini, B.M. Savall, K.D. Carlson, J.M. Williams, M.W. Lanthrop, K.R. Lykke, D.H. Parker, P. Wurz, M.J. Pellin, D.M. Gruen, U. Welp, W.-K. Kwok, S. Fleshler and G.W. Crabtree, J.E. Schirber and D.L. Overmyer, Inorg. Chem., 30, 2962 (1991) 38 V.N. Bezmelnitsyn, A.A. Dityat'ev, V.Y. Davydov, N.G. Shepetov, A.V. Eletskii and V.F. Sinyanskii, Chem. Phys. Lett., 237, 246 (1995) 39 J.A. Schlueter, H.H. Wang, M.W. Lathrop, U. Geiser, K.D. Carlson, J.D. Dudek, G.A. Yaconi and J.M. Williams, Chem. Mater., 5, 720 (1993) 40 R.E. Douthwaite, A.R. Brough and M.L. Green, Chem. Commun., 267 (1994) 41 X. Liu, W.C. Wan, S.M. Owens and W.E. Broderick, J. Am. Chem. Soc., 116, 5489 (1994) 42 J. Chen, Z.-E. Huang, R.F. Cai, Q.-F. Shao, H.J. Ye, Solid St. Commun., 95, 233 (1995) 43 J. Chen, R.F. Cai, Z.-E. Huang, Q.-F. Shao, S.-M. Chen, Solid St. Commun., 95, 239 (1995) 44 P. Bhyrappa, P. Paul, J. Stinchcombe, P.D.W. Boyd and C.A. Reed, J. Am. Chem. Soc., 115, 11004 (1993) 45 P.D.W. Boyd, P. Bhyrappa, P. Paul, J. Stinchcombe, R.D. Bolskar, Y. Sun and C.A. Reed, J. Am. Chem. Soc., 117, 2907 (1995) 46 A. Pénicaud, J. Hsu and C.A. Reed, J. Am. Chem. Soc., 113, 6698 (1991) 47 J. Stinchcombe, A. Pénicaud, P. Bhyrappa, P.D.W. Boyd and C.A. Reed, J. Am. Chem. Soc., 115, 5212 (1993) 48 Y. Chabre, D. Djurado, M. Armand, W.R. Romanow, N. Coutel, J.P. McCauley Jr., J.E. Fischer and A.B. Smith III, J. Am. Chem. Soc., 114, 764 (1992) 49 D. Billaud, S. Lemont and J. Ghanbaja, Synthetic Metals, 70, 1371 (1995) 50 H. Moriyama, H. Kobayashi, A. Kobayashi and T. Watanabe, 115, 1185 (1993) 51 H. Kobayashi, H. Moriyama, A. Kobayashi and T. Watanabe, Synthetic Metals, 70, 1451 (1995) 52 C.A. Foss Jr., D.L. Feldheim, D.R. Lawson, P.K. Dorhout, C.M. Elliott, C.R. Martin and B.A. Parkinson, J. Electrochem. Soc., 140, L84 (1993) 53 U. Bilow and M. Jansen, Chem. Commun., 403 (1994) 54 H. Kobayashi, H. Tomita, H. Moriyama, A. Kobayashi and T. Watanabe, J. Am. Chem. Soc., 116, 3153 (1994) 55 C.J. Carlile, R. Durand, W.K. Fullagar, P.A. Reynolds, F. Trouw and J.W. White, Mol. Phys., 86, 19 (1995) 56 J.T. Masiakowski, M. Puri and L. Kevan, J. Phys. Chem., 95, 1393 (1991) 57 A.A. Zakhidov, A.A. Ugawa, K. Imaeda, K. Yakushi, H. Inokuchi, K. Kikuchi, I. Ikemoto, S. Suzuki and Y. Achiba, Solid St. Commun., 79, 939 (1991)

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3-24 4. Fulleride Structure Determination

In this chapter we describe the crystallographic analysis of the more highly crystalline samples prepared in § 3. The structure elucidation of ammoniated fullerides, including the characterization of several new phases, is detailed in § 4.1, while in § 4.2 we focus on the materials obtained by subsequent deintercalation of ammonia. The final section, § 4.3, is devoted to an examination of diffraction patterns from the materials studied by us on the Time of Flight Crystal Analyser (TFXA) spectrometer at ISIS (see § 6).

Two publicly available1 software packages, PROSZKI2 and GSAS3, were used extensively in what follows. PROSZKI contains four subroutines for the autoindexing of diffraction lines, each employing a different algorithm. Once the basic symmetry has been established, other subroutines may be used to determine the type of lattice, as well as refining the lattice parameters discovered in the first step.

GSAS was used for the Rietveld refinement of X-ray and neutron powder diffraction data. We describe the quality of the fit in terms of the "R-factors", defined as å I-I R= oc, (4.1) p å I o where Io and Ic are observed and calculated intensities, respectively, and

2 å wI()− I wR = oc, (4.2) p å 2 wI o where w are weights derived from an error propagation scheme used in the program.

Where we compare diffraction patterns recorded at different wavelengths, we use a momentum transfer (Q) scale. Q is related to λ and θ by: 4π θ Q= λ sin . (4.3)

4-1 4.1 Fulleride Ammoniates

The liquid ammonia preparation of fullerides described in § 3.1.3 and § 3.1.4 generally led to the incorporation of ammonia in the crystal lattice. The "bomb" synthesis of § 3.1.4 usually led to highly crystalline phases, Table 4.1 summarizing the samples prepared by this technique and their characterization by a variety of techniques.

Table 4.1: Stoichiometries of the "Bomb" Products

NH3 desorption. + X-ray diffraction Planned Alkali AAS* (major phase) NIR Spectrum 3- 2- Li3C60 Li1.3C60(NH3)7.8 Li2C60(NH3)8 ? C60, trace C60 2- 3- Na2C60 Na1.6C60(NH3)6.3 Na2C60(NH3)8 C60, trace C60

Na3C60 Na2.5C60(NH3)6.2 Na3C60(NH3)6 --

Rb2C60 -- (Rb4C60(NH3)2)-- 4- Rb3C60 -- -- C60

Rb4C60 -- Rb4C60(NH3)2 --

Rb5C60 -- (Rb4C60(NH3)2)--

* The indicated stoichiometry was obtained by iterating microanalytic Atomic Absorption Spectroscopy (AAS) results for the alkali metal with the ammonia desorption results, since in the former the ammonia content is unknown, whilst in the latter the alkali metal content is unknown.

In what follows we describe structure determination of the body-centred cubic

Li3C60(NH3)4 and Na3C60(NH3)6 phases (the Li3C60(NH3)4 was prepared by titration), 4 hexagonal Li~2C60(NH3)~8 and Na2C60(NH3)8 , and finally the RbxC60(NH3)y system.

4.1.1 Li3C60(NH3)4

The diffraction pattern of Li3C60(NH3)4 (prepared by titration, see § 3.1.3) was indexed to a bcc lattice. Table 2.2 lists the structural parameters associated with the fit shown in

Figure 4.1. Data for the Li3C60(NH3)4 phase were collected on a HUBER diffractometer in Debye-Scherrer geometry with graphite-analysed CuKα1 radiation. The Li:C60 ratio in the Li3C60(NH3)4 sample was assumed to be 3:1 on the basis of the final absorption

3- spectrum observed in the titration (C60 ; see Table 4.1) - sensible lithium population refinements were not possible with the available diffraction data. Further details of the procedure used to fit the data may be found in reference [5].

4-2 Table 4.2: Refined parameters for Li3C60(NH3)4 Preparative route: liquid ammonia titration Data collection: HUBER diffractometer (see text), λ = 1.5418 Å Refined stoichiometry: Li2.8C60(NH3)3.7 Space group Im 3m a = 11.843 Å wRp = 9.0% Rp = 6.6% Atom x y z U (Å) Frac. Symmetry C 0 .0625(3) .292(1) .070(6) .5 48j C .0588(2) .1904(6) .221(1) .070(6) .5 96l C .1083(5) .1113(5) .255(1) .070(6) .5 96l Li 0 .349(8) .349(8) .01(39) .25 24h N 0 .230(6) .5 .11(2) .64(1) 24g

Figure 4.1: Rietveld fit to Li2.8C60(NH3)3.7. The inset shows details at the higher angles.

4.1.2 Na3C60(NH3)6

Fitting to synchrotron diffraction data from the Na3C60(NH3)6 sample began with the centreing of a merohedrally disordered C60 at 0,0,0, which gave an approximate fit to the lower angle data (2θ ≈ 5° - 40°). Trial and error refinement of the Na populations indicated roughly equal electron densities at 0.5,0,~0.25 and 0.5,~0.14,~0.14, which are

~2.0 Å apart. The Na3C60(NH3)6 phase was assumed to be stoichiometric, ammonia desorption and alkali AAS results (see Table 4.1) being roughly consistent with a

4-3 mixture of Na3C60(NH3)6 and Na2C60(NH3)8, the latter phase (§ 4.1.4) being evident as a minor component in the diffraction pattern. For this reason NaN2 units were anticipated, with Na-N distances of ~2.5 Å9. Regions of electron density were located by placing Na and N atoms in likely sites with appropriate population constraints. Later it was found that elongation of the electron density at 0.5,~0.14,~0.14 along [011] improved the fit. Several alternative arrangements of free Na, NH3 and Na(NH3)x were tried before settling on the arrangement shown in Figure 4.2, which represents a disordered arrangement of NaN2 and Na'N'2 fragments. Refinement of such an arrangement gives Na-N and Na'-N' bond lengths as 2.28 and 2.30 Å, with N-Na-N and N'-Na'-N' bond angles of 164° and 157°, respectively.

Figure 4.2: Pictorial representation of the electron density in a slice through the (0,0.5,0) plane of Na3C60(NH3)6 about the y axis of the unit cell. Occupancies of N and N' sites are twice those of Na and Na' sites, respectively. Adjacent centres of electron density are separated by ~2.0 Å.

Figure 4.3 shows a simplified version of the structure corresponding to the parameters quoted in Table 4.3. The corresponding data fit is shown in Figure 4.4.

We later became aware that this phase had already been characterized in neutron and X-ray diffraction studies by other authors6. While our model agrees in its gross features with the published model, there are differences in that their refinement is performed in the Im 3 space group (ie. their balls are not merohedrally disordered), and they refine only the major orientation of Na(NH3)2 units described in our model. Although their sample was prepared by gas-phase ammonia intercalation, while ours was prepared in

4-4 solution, and while we cannot support our conclusions with neutron diffraction data, it appears that their model leads to rather short Na-N distances - 2.13 and 2.18 Å, compared with the results from our refinement (2.28 and 2.30 Å). The Na-N distances in Na2C60(NH3)8 and Na2CsC60(NH3)4 (2.45 and 2.5 Å respectively) lend support to our model, which also leads to slightly better fits to our synchrotron diffraction data

(wRp = 5.40%, Rp = 3.69%, as opposed to wRp = 5.63%, Rp = 3.99%).

Figure 4.3: Simplified stereo pair of the unit cell of Na3C60(NH3)6, showing Na (green) and N (blue) atoms in their major orientations, in which Na sites are 18% occupied, while N sites are twice this. The refined structure contains merohedrally disordered C60, as well as a minor orientation of disordered Na(NH3)2 units (see text).

Table 4.3: Refined parameters for Na3C60(NH3)6 Preparative route: liquid ammonia bomb Data collection: Big Diff (§ 8.2.3), λ = 1.5486(1) Å Stoichiometry constrained to Na3C60(NH3)6 Refined pattern contains 17.0 wt.% Na2C60(NH3)8 (see Table 4.4) Space group Im 3m a = 12.0845(4) Å wRp = 4.3% Rp = 2.9% Atom x y z U (Å) Frac. Symmetry C 0 .0655 .2904 .020 .5 48j C .0634 .1951 .2137 .020 .5 96l C .1029 .1069 .2510 .020 .5 96l Na' 0 .3952 .3952 .034 .1829 24h N' 0 .2676 .5 .018 .3658 24g H' 0 .2740 .4290 .018 .5488 48j Na 0 .2676 .5 .018 .0671 24g N 0 .3407 .3407 .034 .1342 24h H 0 .3407 .3407 .034 .4025 24h

4-5 Figure 4.4: Rietveld fit to Na3C60(NH3)6. The inset shows the fit to the higher angle data. The lower set of ticks beneath the trace marks peaks due to the Na2C60(NH3)8 phase (see below).

The close similarity of the Li3C60(NH3)4 and Na3C60(NH3)6 structures may be appreciated by comparison of Table 4.2 and Table 4.3; in Li3C60(NH3)4 the relatively poor quality of the data only justified refinement of the major orientation of metal- ammonia fragments seen in Na3C60(NH3)6, with different fractional occupancies of the N atoms accounting for the observed stoichiometry.

It is interesting to compare the structures of Li3C60(NH3)4 and Na3C60(NH3)6 with that of 9 Na2CsC60(NH3)4 . In the latter material, prepared by doping of gaseous ammonia,

Na(NH3)4 tetrahedra are found in the octahedral sites of an fcc array of C60 molecules, the Cs having been forced back into the tetrahedral sites, which then contained 50% uncoordinated Na atoms and 50% Cs atoms. While the preparative methods differ, it had initially been anticipated that the Li3C60 and Na3C60 samples prepared in liquid ammonia would have analogous structures. The lattice parameters for bcc Li3C60(NH3)4 and Na3C60(NH3)6 (11.843 Å and 12.085 Å, respectively) are rather greater than that of 7 bcc Cs3C60 (11.82 Å ), whose hypothetical fcc structure would have an fcc lattice 17 parameter of a ≈ 14.6 Å . The Na2CsC60(NH3)4 fcc lattice parameter (a = 14.473 Å) evidently falls somewhat short of the value where the fcc → bcc structural

4-6 transformation occurs. (In the strontium-doped fulleride, Sr3C60, the bcc phase (a = 11.140 Å) is found to compete with the fcc phase (a = 14.144 Å)8, though note that

6- 3- this compound contains the C60 anion, as opposed to the C60 anion.)

4.1.3 Li~2C60(NH3)~8

Structure determination of this phase has met with considerable difficulty, so that only an outline of the refinement will be presented.

Most peaks in the synchrotron diffraction data from the Li~2C60(NH3)~8 sample may be indexed using a primitive trigonal cell with lattice parameters a = 19.278 Å and c = 10.472 Å. Some additional weak peaks could be indexed by a doubling of this c-axis. Based on typical volumes per C60 molecule in fulleride ammoniates, we expect four C60 molecules per undoubled unit cell in Li~2C60(NH3)~8. However, it does not appear to be possible to place four C60s in such a cell while maintaining reasonable inter-ball distances. On the other hand, three balls are very readily accommodated. Placement of one at 0,0,0 and another at 1/3,2/3,1/2 (in P 3 this generates a third ball at 2/3,1/3,1/2 - see Figure 4.5, which shows the doubled unit cell) led to an approximate intensity fit to the main peaks in the diffraction pattern. With this arrangement, the 3 3 volume per C60 molecule is 1123 Å /C60 - a volume challenged only by the 1159 Å /C60 11 structure of K3C60(NH3)8 tentatively proposed by Zhou et al. .

Figure 4.5: Stereo pair of an approximation of the structure used in the refinement shown in Figure 4.6; the smaller blue balls are N atoms.

4-7 The intensity match for the simple fit is quite convincing, however. Subsequent attempts to locate and refine nitrogen positions as well as ball radii and orientations in the doubled unit cell led to significant improvement of the fit, though concerns regarding the density, the low cell symmetry (and consequently large number of atoms), and the low X-ray scattering power of the unmodelled lithium led to difficulties with the refinement which cannot be readily resolved using the available data.

The structure shown approximately in Figure 4.5, which is the first of its kind among the simple fullerides, leads to the fit shown in Figure 4.6. It does not show obvious simple polyhedra of N atoms such as might be expected to coordinate around Li+, which may reflect the fairly unstable refinements of a model that is in other respects essentially correct. Unit occupancy of the illustrated N sites indicates a C60:NH3 ratio of 1:8, and with such an ammonia content it is tempting to suppose that Li(NH3)4 tetrahedra might 9 be present, analogous to the Na(NH3)4 tetrahedra observed in Na2CsC60(NH3)4 and

Na2C60(NH3)8. This suggests a stoichiometry of Li2C60(NH3)8, in pleasing agreement with ammonia desorption and alkali AAS results (see Table 4.1).

Figure 4.6: Rietveld fit to Li~2C60(NH3)~8. The inset details the deteriorating fit at high angles. The fit has wRp = 10.8%, Rp = 7.8%.

4-8 4.1.4 Na2C60(NH3)8

The alkali ammoniate Na2C60(NH3)8 was modelled in a straightforward manner. The phase was indexed as R-centred trigonal, the cell volume indicating three C60s per unit cell. A C60 was centred at the cell origin in space group R 3; the C60 may be accommodated in this site ( 3 symmetry) by orienting with two opposing hexagons perpendicular to the cell's z axis. The rotation of the C60 is arbitrary in R 3, and an initial value was guessed. Refinement of dummy atom populations located Na roughly, and this was followed by refinement of its z coordinate. Populations of potential N sites 2.4 Å from Na were refined, and two were located giving a roughly tetrahedral 2 coordination. At this stage the C60 was rotated and it was found that χ reached a minimum with C60 in the 3m orientation. A disordered C60 was also tried with 50% of

C60s in the 3m orientation, and the others in various 3 orientations, but again the 3m orientation was definitely preferred. The symmetry was now raised to R 3m, requiring only six unique carbons. These six carbons and the NaN4 unit were then refined alternately, and background profile parameters were added. Hydrogen atoms were added at calculated positions - two alternatives were tried at each site, cis and trans. The preferred arrangement had one site cis, the other trans. The final refinement was made with all C, N and Na refining. Allowing N populations to vary led to fractional occupancies of 0.991(7) and 1.070(14), so that the stoichiometry is clearly

Na2C60(NH3)8.

Table 4.4 lists the results of the fit, which is shown in Figure 4.7, and Figure 4.8 shows the arrangement of atoms in the unit cell. The form of trigonal packing observed in

Na2C60(NH3)8 is unusual, and is perhaps a consequence of the tetrahedral Na(NH3)4 9 units, whose Na-N distance is 2.45 Å (cf. 2.5 Å in Na2CsC60(NH3)4 ). The distance + between adjacent Na ions is 3.93 Å. The volume per C60 molecule in the unit cell 3 (959 Å /C60) is large when compared with other well-characterized fulleride 3 9 3 10 ammoniates (758 Å /C60 for Na2CsC60(NH3)4 , 763 Å /C60 for K3C60(NH3)1 , 3 3 3 807 Å /C60 for Rb4C60(NH3)2, 831 Å /C60 for Li3C60(NH3)4), and 882 Å /C60 for

Na3C60(NH3)6), and suggests that the packing here is influenced by factors other than steric constraints.

4-9 Table 4.4: Refined parameters for Na2C60(NH3)8 Preparative route: liquid ammonia bomb Data collection: Big Diff (§ 8.2.3), λ = 1.5486(1) Å Refined data range 2θ = 8° - 83° 65 variables refined for 7229 data points Refined stoichiometry: Na2C60(NH3)8 Refined pattern contained 20.8 wt.% Na3C60(NH3)6 (see Table 4.3) Space group R 3m a = 12.21647(16) Å c = 22.2673(4) Å wRp = 5.57% Rp = 3.50% AtomxyzUiso Frac. Symmetry C .1401(4) -.1401(4) .0828(4) .0161(2) 1 36i C .2065(6) -.0168(6) .1094(2) .0161(2) 1 36i C .1179(3) .0018(4) .1470(2) .0161(2) 1 36i C .1636(3) -.1636(3) .0255(4) .0161(2) 1 36i C -.0885(4) -.2931(4) .0723(2) .0161(2) 1 36i C -.2546(4) -.3187(5) .0107(3) .0161(2) 1 36i N .1119(2) .2237(4) .4386(2) .073(3) 1 36i H 0.1358 0.2716 .4691(2) 0.08 1 36i H 0.1770 0.2361 .4154(2) 0.08 1 36i Na 0 0 .4117(3) .084(3) 1 6c N 0 0 .3011(4) .073(3) 1 6c H 0.0395 0.0791 .2855(4) 0.08 1 36i

Figure 4.7: Rietveld fit to Na2C60(NH3)8. The inset shows the fit at higher angles. The upper ticks mark peaks due to minor contamination by Na3C60(NH3)6.

4-10 Figure 4.8: Stereo pair of the unit cell of Na2C60(NH3)8, showing the Na(NH3)4 tetrahedra (H atoms not shown).

4.1.5 RbxC60(NH3)y

During the course of the work described here, a report appeared in the literature demonstrating that the reversible doping of Na2CsC60 with gaseous ammonia can enlarge the cubic lattice parameter and result in an increased superconducting Tc, with 9 formation of the compound Na2CsC60(NH3)4 . The same workers mentioned the reaction of Na2RbC60 to form an analogous compound, as well as the reaction of K3C60 with ammonia to yield K3C60(NH3)1, which has since been characterized in separate studies10 11 12. In reference [11] evidence was also found for a more highly ammoniated phase, K3C60(NH3)8. Surprisingly, no reaction was observed with Rb3C60.

The studies presented here initially focused on ammoniation of Rb3C60, this having the 9 highest Tc of the simple binary fullerides. Contrary to Zhou et al. we found that addition of ammonia gas to Rb3C60 led to dramatic changes in the X-ray diffraction pattern (such as may be seen by comparison of Figure 2.5 (Rb3C60) and Figure 4.11

(Rb3C60(NH3)2.5)). The X-ray diffraction pattern changes significantly on annealing at 70 - 100ºC. Material at early stages of annealing and aging can be indexed on, and have correct semi-quantitative peak intensities for, an icosahedral quasicrystalline model5.

This suggested local ordering governed by the C60 molecular symmetry, but large peak widths suggested poor long-range ordering. The majority of peaks in a synchrotron

4-11 powder pattern of a highly annealed and aged sample of Rb3C60(ND3)2.5 can be indexed and refined in a body-centred orthorhombic cell, with other peaks from Rb1C60 and 13 Rb3C60(NH3)4 . The stoichiometry of the new orthorhombic phase is Rb4C60(NH3)2, indicating that ammonia doping has induced disproportionation of the parent Rb3C60.

Synchrotron diffraction data from a purer sample of Rb4C60(NH3)2 prepared by the bomb technique (§ 3.1.4) has since allowed further structure refinement, the results of which are presented in Figure 4.9, Table 4.5 and Figure 4.10.

Figure 4.9: Stereo pair showing atoms in the unit cell of Rb4C60(NH3)2. The crystal symmetry is such that two orientations of the NH3 molecule are crystallographically identical.

The orthorhombic symmetry of the unit cell (Immm) suggests that unlike tetragonal

Rb4C60 (I4/mmm), the C60 molecules in Rb4C60(NH3)2 are orientationally ordered. This was confirmed by attempts to perform refinements using merohedrally disordered C60 molecules. Nevertheless the sample appears to contain a small fraction of deammoniated (normal) Rb4C60, perhaps due to the procedure used to remove the liquid ammonia in the final stage of preparation (distillation in a sealed tube at -78ºC). In the original refinement of neutron diffraction data for the Rb3C60(ND3)2.5 sample it was possible to refine the positions of deuterium atoms, which were used as a guide in the placement of H atoms in the X-ray refinement presented here (relative to Rb, N and C, the neutron scattering length of deuterium is much larger than the relative X-ray scattering power of H).

4-12 Table 4.5: Refined parameters for Rb4C60(NH3)2 Preparative route: bomb Data collection: Big Diff (§ 8.2.3), λ = 1.5486(1) Å Space group Immm Refined phase stoichiometry: Rb3.7C60(NH3)1.8 Refinement contained 8.6 wt.% Rb4C60 a = 12.5925(5) Å, b = 11.6682(4) Å, c = 10.9867(4) Å wRp = 4.14% Rp = 3.01% AtomxyzUiso Frac. Symmetry C .060(2) 0 .320(2) .040(4) 1 8m C .195(2) .0639(9) .208(2) .040(4) 1 16o C .108(2) .108(2) .288(2) .040(4) 1 16o C .049(2) .204(1) .231(2) .040(4) 1 16o C .094(1) .273(2) .117(2) .040(4) 1 16o C 0 .308(2) .062(2) .040(4) 1 8l C .191(1) .218(2) .071(2) .040(4) 1 16o C .254(2) .120(2) .120(2) .040(4) 1 16o C .269 (2) .047(2) 0 .040(4) 1 8n Rb .2241(3) 0.5 0 .039(3) .936(6) 4f Rb 0.5 .1888(3) 0 .017(3) .913(5) 4h N 0 0.5 .285(2) .009(9) .91(2) 4j H 0 0.42 .264(4) .025 .453(7) 8l H 0.08 0.45 .264(4) .025 .453(7) 16o

Figure 4.10: Rietveld fit to Rb4C60(NH3)2 (upper tick marks), containing some Rb4C60 (lower ticks). The inset shows the fit at higher angles.

4-13 As illustrated in the difference between the diffraction profile and the final fit (lower trace in Figure 4.10), it appears that some aspect of the structure may yet be unaccounted for. It is possible that this is due to a degree of alkali and/or ammonia site disordering (cf. the Na3C60(NH3)60 structure already described), a notion supported by the low Rb and N site occupancies in the final fit (Table 4.5). From Figure 4.9 it can be seen that the NH3 molecules are surrounded by an approximately tetrahedral cage of Rb atoms. However, the Rb-N distances shown as purple lines are considerably shorter than those connecting to the other two corners of the tetrahedron, the refined distances being 3.23 Å and 4.21 Å, respectively. The Rb-N-Rb angle is 85.8º.

The observation of disproportionation of Rb3C60 upon addition of ammonia was exciting in the context of superconductivity, a propensity for disproportionation being common among superconducting compounds14 (see also § 2.3). Disproportionation has been observed in other fullerides also15 16 17, though it is not clear whether such phenomena arise as the result of structural or electronic instabilities (see § 2.2).

To further investigate the ammonia-induced disproportionation of Rb3C60, a series of samples with nominal composition RbxC60(NH3)y (x = 2, 3, 4 and 5, various y) were prepared in liquid ammonia, diffraction patterns of which are shown in Figure 4.11.

The presence of the Rb4C60(NH3)2 phase in all the samples is clear, indicating that the

2- 3- 5- C60, C60 and C60 anions originally present in the solutions have all disproportionated in

4- the solid state so as to give the product containing the C60 anion. This result suggests that the Rb4C60(NH3)2 phase may have an energy low enough to drive the necessary disproportionation reactions, rather than that the various anions are intrinsically prone to disproportionation. This supposition is supported by the even spacing (~0.46 V) of the 18 first six reduction potentials of C60 .

The other products formed in the disproportionation reactions are not so easily assigned, except in the case of the "Rb5C60(NH3)y" material, where it is clearly Rb6C60 (compare

Figure 2.5). The "Rb3C60(NH3)2.5" material has been described by us in reference [13], where two other phases (polymeric Rb1C60 and very poorly crystalline Rb3C60(NH3)4) have been found in simultaneous fits to neutron and X-ray diffraction data. Other phases in the "Rb2C60(NH3)y" material have proven difficult to model, due partly to the

4-14 1- large number of potentially oligomeric phases that are possible for salts of the C60 anion (see § 2.1.5).

The structural similarity of the Rb4C60(NH3)2 and Rb6C60 phases is quite striking - both are merohedrally ordered, and in Rb4C60(NH3)2 the ammonia molecules take the place of the additional Rb atoms in Rb6C60, with some cell symmetry lowering. The Rb4C60 phase is also structurally closely related to Rb4C60(NH3)2, being generated from it by removal of the NH3 molecules and merohedrally disordering the C60s so as to equate the orthorhombic a and b axes and raise the cell symmetry to tetragonal (see Figure 2.4).

Rb5C60 (NH3) y

Rb4C60 (NH3) 2 Intensity

Rb3C60 (NH3) 2.5

Rb2C60 (NH3) y

0.5 1.0 1.5 2.0 2.5 3.0

-1 Q =(4 π/ λ)sin θ (Å )

Figure 4.11: Synchrotron diffraction patterns from RbxC60(NH3)y samples prepared in liquid ammonia.

4-15 4.2 Fullerides Prepared by Deammoniation

By the removal of ammonia from the products prepared in liquid ammonia, a number of fullerides were prepared whose structural characterization is described here.

4.2.1 Lithium and Sodium Fullerides

Deammoniation and annealing of ammonia-containing Li~2C60(NH3)~8, Li3C60(NH3)4,

Na2C60(NH3)8 and Na3C60(NH3)6 samples (§ 4.1) led to the diffraction patterns shown in

Figure 4.12. Unlike most other fullerides, deammoniated Li2C60 is pyrophoric, and caught fire when the X-ray capillary was accidentally broken after the diffraction measurement. This observation confirms our belief that samples had not suffered significant air exposure prior to or during the X-ray diffraction measurement.

At the time of writing there are only a handful of papers describing the Li:C60 binary 19 20 21 system (ternary Li:A:C60 structures are summarily described in § 2.1.1), these describing synthetic conditions and sample characterization quite different from our own. Literature pertaining to the Na:C60 system is much more substantial, but also suggests it is poorly understood. Douthwaite et al.22 observed NMR signals due to equimolar quantities of C60 and Na2C60 in a sample prepared by heating the two together at 600ºC for several days, demonstrating that intermediate phases are not formed. This result would seem to cast some doubt upon the supposition by Bezmelnitsyn and 23 coworkers that Na1C60 is formed upon reaction of sodium with C60 in toluene solution. 24 On the other hand, Yildirim et al. describe the phase separation of NaxC60 into C60 and 22 Na1C60 for 0 < x < 1 at temperatures up to 500ºC (syntheses lasting ~12 days ) with solid solutions forming for 1 < x < 3. Below 250 K there is some evidence for the 22 disproportionation of Na3C60 into Na2C60 and Na6C60 .

Our diffraction patterns from Li2C60, Li3C60, Na2C60 and Na3C60 prepared by deammoniation are mutually similar despite their very different annealing times (Table

4.6), and also bear a striking resemblance to C70 (bottom trace in Figure 4.12, obtained

4-16 Li2C60

Li3C60

Intensity Na2C60

Na3C60

C70

0.5 1.0 1.5 2.0 2.5 3.0

-1 Q = (4 π/ λ)sin θ (Å ) Figure 4.12: Products from liquid ammonia preparations after deammoniation and annealing under the conditions shown in Table 4.6. A diffraction pattern of C70 is shown for comparison.

Table 4.6: Deammoniation conditions Sample Preparation Deammoniation Anneal Li~2C60(NH3)~8 bomb 10 min at 300ºC 4 days at 300ºC Li3C60(NH3)4 titration 1 hour at 320ºC (none) Na2C60(NH3)8 bomb 10 min at 300ºC 11 days at 300ºC Na3C60(NH3)6 bomb 10 min at 310ºC 11 days at 300ºC

4-17 by heating the product precipitated from toluene to 300°C for 12 hours under diffusion -5 pump vacuum (~10 τ)). Analysis of similar C70 patterns by other authors has shown that samples prepared in this way contain mainly hcp molecules with ~5% stacking faults25, which suggests that when prepared by deammoniation, the Li and Na fullerides are similarly faulted.

The deammoniated Na2C60 sample differs from the TFXA Na2C60 sample described in

§ 4.3.1 (which was prepared by heating C60 and sodium metal at 340ºC for 5 days) in that when dissolved in toluene, it did not appear to contain any undoped C60, perhaps testifying to the homogeneity of the product produced by our solution-phase synthesis.

The TFXA sample also shows some evidence for C60 stacking faults (see Figure 4.14 in § 4.3.1), though the relevant features (in particular the satellite peaks on either side of the strong peak at Q ≈ 0.77 Å-1) are much less prominent.

Taken together, these results indicate that our deammoniation/annealing synthesis is not suited to the preparation of Pa 3 Na2C60 (described in § 2.1.1), but also that this structure may not constitute a particularly deep energy minimum, with the structure of the product being dictated in large measure by the synthetic approach. The latter supposition seems consistent with the general confusion in the literature concerning the NaxC60 (0 < x < 3) system. The similarity of the patterns in Figure 4.12 indicates that the same conclusion may well apply to the Li:C60 system.

Difficulties in modelling possible stacking faults are compounded by the possibility of orientational ordering of the balls. Nevertheless, an attempt was made to fit the Li2C60 data to two phases, one containing an hcp array of balls, the other an fcc array. The

Li2C60 data were chosen for the fit because of the negligible X-ray scattering power from lithium.

The fcc phase used in the model was merohedrally disordered, C60 atomic coordinates 30 being appropriate to K3C60 , but allowed radial refinement. The lattice was refined to

13.981 Å. For comparison the lattice parameters of Li2CsC60 and Li2RbC60 are 13.998 Å and 13.896 Å, respectively26. The hexagonal phase was modelled in space group P3, with one C60 at 0,0,0, the other at 1/3,2/3,1/2. Lattice refinements led to a = 9.857 Å, c = 16.356 Å, corresponding to cubic parameters of 13.94 Å and 14.17 Å

4-18 respectively. Given the poor crystal quality and reason to suspect stacking faults, a and c were adjusted to give an average cubic lattice parameter of 14.015 Å (not significantly different from the value used for the fcc phase), and fixed at this value. The balls were then allowed radial refinement. Refinement of peak shapes of both phases was unstable, the associated parameters finally being fixed at values which led to reasonable visual fits. The background was modelled as a 6-term shifted Chebyshev function.

The fit in Figure 4.13 suggests that the material is predominantly hcp, the ratio of the hcp to fcc phases being approximately 6:4.

Figure 4.13: An attempted fit of the deammoniated Li2C60 sample to fcc (upper ticks) and hcp (lower ticks) arrangements of C60 molecules (see text).

4.2.2 Rubidium Fullerides

Deammoniation and annealing at 320 - 350ºC of liquid ammonia preparations of

Rb3C60(NH3)2.5 and Rb4C60(NH3)2 for 3 hours and 3 weeks (respectively) led to diffraction signatures of Rb3C60 and Rb4C60, with the reappearance of superconductivity in the former material27, which had been disproportionated by the ammonia (see § 4.1.5

4-19 and reference [13]). Rb3C60 and Rb4C60 samples prepared in this way were particularly phase-pure and gave sharp diffraction peaks, and were used in the compilation in Figure

2.5 (§ 2.1.2). This is in contrast with our Rb3C60 and Rb4C60 samples prepared by the vapour-phase method (§ 3.1.1) which were generally not phase-pure, and also with the lithium and sodium fullerides described in § 4.2.1, whose diffraction patterns were indicative of considerable disorder.

4.3 Na2C60 and Rb4C60 TFXA Samples

In connection with our studies of the fulleride vibrational spectroscopy, samples of nominal composition Na2C60 and Rb4C60 were prepared by vapour-phase intercalation (§ 3.1.1) for inelastic neutron scattering (INS) measurements on the Time of Flight Crystal Analyser (TFXA) spectrometer at ISIS (§ 6). The characterization of these samples by least squares fitting their X-ray diffraction patterns is described here with reference to the structures described in the literature28 29. The TFXA instrument's neutron diffraction detector, while not capable of producing data of sufficient quality for refinements, was useful as an in-situ check of the approximate stoichiometry, and gave the lattice parameters at low temperatures.

4.3.1 Na2C60

Figure 4.14 compares the synchrotron X-ray diffraction pattern of the TFXA Na2C60 sample with diffraction patterns of C60 and Na2C60 obtained by deammoniating and annealing the product obtained in liquid ammonia as described further in § 4.2. X-ray diffraction patterns of the Na2C60 sample recorded a month prior to the TFXA experiment and five months afterwards were identical. Most peaks are readily indexed as primitive cubic with a lattice parameter (a = 14.194 Å) appropriate to an fcc array of orientationally ordered balls, as expected for Pa 3 C60. At the temperature of the TFXA measurements (9 K), the lattice had contracted to 14.14 Å. In addition to the primitive cubic peaks, several other peaks may be seen which could not be indexed either as

4-20 C60 Intensity TFXA Na2C60

Bomb Na 2C60

0.5 1.0 1.5 2.0 2.5 3.0

-1 Q = (4 π/ λ)sin θ (Å )

Figure 4.14: Comparison of the X-ray diffraction pattern of the Na2C60 sample prepared for TFXA experiments with C60 and a sample of Na2C60 prepared by deammoniation and annealing at 340ºC for 11 hours. supercells of the primitive cubic phase or collectively as a new phase. The weak satellite peaks on either side of the strong cubic peak at Q ≈ 0.77 Å-1 appear in published 22 24 28 diffraction patterns of sodium-doped C60 ; reference [24] ascribes them to a highly faulted hexagonal close-packed (hcp) minority phase, an interpretation supported by their observation in the deammoniated and annealed Na2C60 bomb product (§ 4.2.1). Other peaks may be seen in Figure 4.14, notably at Q ≈ 1.07, 1.13, 1.33, 1.41 and 1.50 Å-1, which are not due to an hcp minority phase. None of these can be attributed to unreacted sodium, or sodium oxides, sodium carbonate or sodium oxalate, which could conceivably have been formed had air leaked on to the sample. The extra peaks also do not correspond to any observed in the sodium heterocluster compounds described by Yildirim et al.28, and while they have defied attempts at indexing, the associated d-spacings are large enough to suggest that the relevant contaminants are likely to include C60. Where possible, such peaks were excluded from the fits described below. For these reasons, and owing to the difficulty of modelling the orientationally disordered

4-21 undoped C60 also present in the sample, it was difficult to obtain a good fit to the synchrotron diffraction data.

The overall stoichiometry of the sample was consistent with atomic absorption spectroscopy (AAS) results, which indicate a sample stoichiometry of Na1.97C60.

As can be seen from Figure 4.14, C60 and Na2C60 have similar structures with almost identical lattice constants. A consequence of this is that estimation of the relative proportions of the two phases in a given sample is difficult by diffraction methods. The fits depend crucially on the different peak intensity ratios, which in turn depend on how well these are modelled.

The undoped C60 component of the sample was modelled using the coordinates from 30 merohedrally disordered K3C60 , with refinement of atomic positions and anisotropic thermal parameters in order to mimic the known orientationally disordered C60 31 structure . In this way it was possible to produce a good fit to C60 powder diffraction data obtained using a HUBER diffractometer in Debye-Scherrer geometry with graphite- analysed CuKα1 radiation (upper trace in Figure 4.14). A stoichiometric Pa 3 Na2C60 phase was entered, using the coordinates for a C60 molecule generated by a macro included with the PC version of the GSAS suite. The balls were initially in one of the merohedral orientations, and were allowed radial refinement prior to stepwise rotation about the [111] axes as shown in Figure 2.2 (§ 2.1.1). In the fit to each orientation, five iterations were performed in which 12 Chebyshev background coefficients, phase fractions of the Na2C60 and undoped C60, and their lattice parameters were simultaneously refined. In this way Figure 4.15 was produced, which closely mirrors that observed in reference [24] and also the bottom panel of Figure 2.3.

A procedure similar to that used in reference [24] was then used to improve the fit. In summary, three different orientational phases were entered, corresponding to the three minima seen in Figure 4.15, and the phase fractions were refined with the constraint that if the phase fraction in the deep (Γ = 22º) minimum is kp, then the fraction in both the local (Γ = 60º and Γ = 105º) minima is k(1 - p)/2 for some value of k (k is unity if the sample contained no undoped C60 impurity).

4-22 0.2 0.145

0.14 wRp 0.19 Rp 0.135

0.18 0.13 Rp wRp 0.125 0.17

0.12 0.16 0.115

0.15 0.11 5 15 25 35 45 55 65 75 85 95 105 115 Rotation Angle (degrees)

Figure 4.15: R-factors for fits to orientationally disordered C60 and Na2C60 as a function of the rotation angle Γ (see § 2.1.1).

The final refinement is shown in Figure 4.16, and has wRp = 11.17% and Rp = 8.15%, the weight fraction of undoped C60 being 18.1(4)% and the value of p being 0.72(2) (note that the latter errors do not take into account any inaccuracies in the modelling of

C60 or the probable superposition of some of the unknown impurity peaks on the refined peaks in the diffraction histogram, and must therefore be treated as a potentially poor approximation). For comparison, Yildirim et al.24 obtain p ≈ 0.62 in a sample that appears to be rather more phase pure. Further refinement of sodium populations failed to converge to reasonable values.

Further evidence for undoped C60 was found by other techniques; the intramolecular -1 INS results for Na2C60 (§ 6.2.4) show a clear peak at 425 cm (the Hg(2) mode) which is present in undoped C60 but greatly dispersed in the other incompletely doped fullerides.

More significantly, a small portion of the Na2C60 sample, when partially dissolved in dry degassed toluene gave a strong magenta colour, as is observed for C60. That this was not due to solvent-induced disproportionation (as has been observed elsewhere in this work, see § 4.1.5) was confirmed by attempting to dissolve a sample of deammoniated and annealed Na2C60 prepared in liquid ammonia in the same dry degassed toluene. This showed no magenta colouration even after several hours of intermittent and vigorous agitation. Admitting air initially did not change the situation, but after overnight standing in contact with air, a strong magenta solution was observed.

4-23 Figure 4.16: The final fit to the TFXA Na2C60 diffraction pattern. The bottom three sets of tick marks beneath the trace represent Na2C60 with C60 in the orientations corresponding to minima in Figure 4.15; the upper set marks the undoped C60 peaks.

An attempt was made to quantify this observation by appending a quartz cuvette to the apparatus. After about 40 minutes of vigorous agitation and occasional ultrasonication, comparison of the absorbance with a standard C60 solution in the same cuvette indicated that the TFXA sample contained 10% undoped C60 by weight. After overnight standing under vacuum, the optical absorbance indicated 27 wt.% C60. This and the X-ray refinement result (~18 wt.% C60, with other unknown impurities) demonstrate that considerable uncertainty remains as to the detailed composition of the sample.

4.3.2 Rb4C60

Figure 4.17 shows a fit of the X-ray diffraction pattern of the nominal Rb4C60 TFXA 29 sample to the known phases Rb3C60, Rb4C60 and Rb6C60 . Diffraction patterns of the pure constituent phases have been compared in Figure 2.5 (§ 2.1.2). Here the phases have been assumed to be stoichiometric. The synchrotron X-ray diffraction pattern, for which higher angle data are unavailable, was recorded 5 months after the inelastic

4-24 Figure 4.17: fit to the Rb4C60 TFXA sample's X-ray diffraction pattern (wRp = 5.92%, Rp = 4.30%); Rb sites are assumed completely filled. Ticks beneath the pattern mark the peaks of (top to bottom) Rb6C60, Rb4C60 and Rb3C60. The overall stoichiometry is Rb3.67C60.

Figure 4.18: same as Figure 4.17, but Rb occupancies were refined also (see text), leading to wRp = 4.94%, Rp = 3.69%. The overall stoichiometry is Rb3.39C60.

4-25 neutron scattering measurements. The lattice parameter of Rb3C60 (Fm 3m) refines to a = 14.422(1) Å, those of Rb4C60 (I4/mmm) are a = 11.959(1) Å and c = 11.014(1) Å, 29 and for Rb6C60 (Im 3) it is a = 11.540(5) Å, in good agreement with published values .

Rb6C60 was present in such small quantity that only the phase fraction and lattice parameter were refined; for the other two phases the positional and thermal parameters were also refined. The fit suggests an overall stoichiometry of Rb3.67C60, reasonably close to the target stoichiometry for the synthesis. The lower trace in the figure, showing the difference between the observed and calculated intensities, reveals the considerable difficulty that was had in modelling peak shapes - attempts to refine these in various ways while holding all other parameters fixed did not lead to significant improvement. Similar difficulties have been described by other authors32. This fact, as well as the presence of three phases, suggest that the preparative conditions described in § 3.1.1 produce a sample that is considerably disordered.

Although it is believed that as usually prepared, Rb3C60, Rb4C60 and Rb6C60 phases are close to stoichiometric32, another fit to the TFXA sample's diffraction data was performed in which the Rb occupancies were refined. This is shown in Figure 4.18. As suggested in the captions to Figure 4.17 and Figure 4.18, the latter fit is significantly better in terms of the associated R factors, and suggests an overall stoichiometry of only

Rb3.39C60. The Rb occupancy in all phases refines to between 0.85 and 1.00. On the other hand, differences between the observed and calculated patterns suggest that the low occupancies may yet be an artefact of the poor fitting of the peak shapes.

The neutron diffraction results taken in situ on TFXA indicated the Rb3C60 and Rb4C60 lattices contract to a = 14.35 Å and a = 11.93 Å, c = 10.89 Å, respectively at 9 K. No

Rb6C60 diffraction peaks were strong enough to be observed in the neutron diffraction patterns. The relative neutron diffraction peak areas suggest an overall stoichiometry of

Rb3.68C60, in good agreement with the stoichiometry deduced from the fit in Figure 4.17.

Several attempts were made to refine the carbon positions on this and other vapour- phase intercalation preparations. While such refinements can be made to lead to much improved fits to the data, they generally involve the atoms moving to unreasonable positions on the surface of the C60 sphere, suggesting unmodelled orientational disorder.

4-26 4.4 References

1 At the time of writing, the relevant software can be downloaded from the ftp site: ftp.minerals.csiro.au/pub/xtallography. 2 W. Lasocha and K. Lewinski, Proszki System for Powder Data Processing, Department of Crystal Chemistry and Crystal Physics, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland 3 A.C. Larson and R.B. von Dreele, 1985-1990, General Structure Analysis System, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 4 W.K. Fullagar, P.A. Reynolds and J.W. White, (submitted to Solid State Communications, December 1996) 5 R. Durand, W.K. Fullagar, G. Lindsell, P.A. Reynolds and J.W. White, Mol. Phys., 86, 1 (1995) 6 P.F. Henry, M.J. Rosseinsky and C.J. Watt, J. Chem. Soc., Chem. Commun., 2131 (1995) 7 A. Messaoudi, J. Conard, R. Setton and F. Béguin, Chem. Phys. Lett., 202, 506 (1993) 8 A.R. Kortan, N. Kopylov, E. Özdas, A.P. Ramirez, R.M. Fleming and R.C. Haddon, Chem. Phys. Lett., 223, 501 (1994) 9 O. Zhou, R.M. Fleming, D.W. Murphy, M.J. Rosseinsky, A.P. Ramirez, R.B. van Dover and R.C. Haddon, Nature, 362, 433 (1993) 10 M.J. Rosseinsky, D.W. Murphy, R.M. Fleming and O. Zhou, Nature, 364, 425 (1993) 11 O. Zhou, T.T.M. Palstra, Y. Iwasa, R.M. Fleming, A.F. Hebard, P.E. Sulewski, D.W. Murphy and B.R. Zegarski, Phys. Rev. B, 52, 483 (1995) 12 Y. Iwasa, H. Shimoda, T.T.M. Palstra, Y. Maniwa, O. Zhou and T. Mitani, Phys. Rev. B, 53, R8836 (1996) 13 W.K. Fullagar, D. Cookson, J.W. Richardson Jr., P.A. Reynolds and J.W. White, Chem. Phys. Lett, 245, 102 (1995) 14 A.W. Sleight, Acc. Chem. Res., 28, 103 (1995) 15 G. Figel, G. Bortel, M. Tegze, L. Granasy, S. Pekker, G. Oszlanyi, O. Chauvet, G. Baumgartner, L. Forro, P.W. Stephens, G. Mihaly and A. Janossy, Phys. Rev. B, 52, 3199 (1995) 16 T. Enoki, Y. Ohtsu, K. Suzuki, K. Imaeda, A.A. Zakhidov, K. Yakushi, K. Kikuchi, S. Suzuki and Y. Achiba, Synthetic Metals, 64, 329 (1994) 17 M.J. Rosseinsky, D.W. Murphy, R.M. Fleming, R. Tycko, A.P. Ramirez, T. Siegrist, G. Dabbagh and S.E. Barrett, Nature 356, 416 (1992) 18 Q. Xie, E. Pérez-Cordero and L. Echegoyen, J. Am. Chem. Soc. 114, 3978 (1992) 19 C. Gu, F. Stepniak, D.M. Poirier, M.B. Jost, P.J. Benning, Y. Chen, T.R. Ohno, J.L. Martins, J.H. Weaver, J. Fure and R.E. Smalley, Phys.Rev. B, 45, 6348 (1992) 20 Y. Chabre, D. Djurado, M. Armand, W.R. Romanow, N. Coustel, J.P. McCauley Jr., J.E. Fischer and A.B. Smith III, J. Am. Chem. Soc., 114, 764 (1992) 21 D. Billaud, S. Lemont and J. Ghanbaja, Synthetic Metals, 70, 1371 (1995) 22 R.E. Douthwaite, M.L.H. Green, M.J. Rosseinsky, Chem. Mater., 8, 394 (1996) 23 V.N. Bezmelnitsyn, A.A. Dityat'ev, V.Y. Davydov, N.G. Shepetov, A.V. Eletskii and V.F. Sinyanskii, Chem. Phys. Lett., 237, 246 (1995) 24 T. Yildirim, J.E. Fischer, A.B. Harris, P.W. Stephens, D. Liu, L. Brard, R.M. Strongin and A.B. Smith III, Phys. Rev. Lett., 71, 1383 (1993) 25 M.C. Valsakumar, N. Subramanian, M. Yousuf, P.C. Sahu, Y. Harihan, A. Bharathi, V. Sankara Sastry, J. Janaki, G.V.N Rao, T.S. Radhakrishnan and C.S. Sundar, Phys. Rev. B, 48, 9080 (1993) 26 K. Tanigaki, T.W. Ebbesen, J.S. Tsai, I. Hirosawa and J. Mizuki, Europhys. Lett., 23, 57 (1993)

4-27 27 C.J. Carlile, R. Durand, W.K. Fullagar, P.A. Reynolds, F. Trouw and J.W. White, Mol. Phys., 86, 19 (1995) 28 T. Yildirim, O. Zhou, J.E. Fischer, N. Bykovetz, R.A. Strongin, M.A. Cichy, A.B. Smith III, C.L. Lin and R. Jelinek, Nature, 360, 568 (1992) 29 P.W. Stephens, L. Mihaly, J.B. Wiley, S-M. Huang, R.B. Kaner, F. Diederich, R.L. Whetten and K. Holczer, Phys. Rev. B, 45, 543 (1992) 30 K.M. Allen, W.I.F. David, J.M. Fox, R.M. Ibberson and M.J. Rosseinsky, Chem. Mater., 7, 764 (1995) 31 P.C. Chow, X. Jiang, G. Reiter, P. Wochner, S.C. Moss, J.D. Axe, J.C. Hanson, R.K. McMullan, R.L. Meng and C.W. Chu, Phys. Rev. Lett., 69, 2943 (1992) 32 J.E. Fischer, G. Bendele, R. Dinnebier, P.W. Stephens, C.L.Lin, N. Bykovetz and Q. Zhu, J. Phys. Chem. Sol., 56, 1445 (1995)

4-28 5. Molecular Electronics

In § 2.2 we described how the molecular orbitals of the C60 anions form bands when in the solid state, and how this determines the electronic properties of the fulleride salts. In

n- this section we seek a better understanding of the electronic structure of the isolated C60 (n = 1 - 6) anions by the examination of their infrared absorption spectra, which were obtained by spectrophotometric titration in § 3.1.3, and by an electrosynthetic technique we have developed.

5.1 Background

1 A Hückel molecular orbital diagram of C60 was calculated by Hale and soon after improved by Brus2. The predicted ordering of the frontier orbitals is reproduced in Figure 5.1, and the frontier orbital transitions relevant to the current work are marked.

n- In the C60 (n = 1 - 6) anions, which are the species that primarily concerns us, the triply degenerate t1u lowest unoccupied molecular orbitals (LUMOs) are of particular interest.

(Superconducting salts of C60 with the alkaline earth metals have also been synthesized, in which it is known that the LUMOs are the higher energy t1g orbitals.)

Experimental studies of the higher anions of C60 are difficult owing to their very negative reduction potentials3 4. Nevertheless, there have been a number of attempts in

n- the literature to account for the C60 (n = 1 - 6) anions' transitions in the range 7000 - 16000 cm-1 (near-infrared/visible, NIR/vis) and electron spin resonance (ESR) spectra.

1- 5 6 7 Many of these focus on the C60 anion , this being the easiest to generate and probably the simplest to understand, since the presence of only one excess electron means there can be no interelectronic effects in the valence orbitals. For example, only the monoanion absorption spectrum has been carefully measured in inert gas matrices8 9, though less well resolved spectra have been obtained by a number of workers by chemical or electrochemical reduction in a variety of solvents. Even with this anion, however, there appears to have been some confusion in the literature, caused by sample

5-1 2- contamination (what appears to be a low concentration of C60 in spectra obtained by 10 11 γ-irradiation of a glassed solution of C60 led to the conclusion that a polar matrix

1- 12 causes a lowering of symmetry in the C60 anion ).

Figure 5.1: Frontier orbitals of C60. In the anions the t1u level begins to fill. The illustrated approximate energies (in cm-1) are deduced from the observed optically allowed (green) and forbidden (red) transitions seen in Figure 5.2; values in parenthesis are from Heath et al.13.

14 2- Boyd et al. give a rigorous experimental account of the electronic situation in the C60 anion by collating data obtained using a variety of techniques. They conclude that the ion is paramagnetic, but that triplet and singlet electronic ground states are too close in energy to be able to determine which has lower energy (throughout this section the terms "singlet", "doublet", etc. refer to spin states).

3- Frontier orbital splitting patterns which could perhaps account for the observed C60 and

4- 15 C60 NIR/vis spectra have been described by Lawson et al. . As stated by these workers,

3- the C60 anion spectrum is unexpected in that the exact half-filling of the t1u orbitals gives no reason to suppose that Jahn-Teller (JT) lowering of the t1u orbitals' degeneracy will occur, and from Hund's rule we would expect a quartet ground state. However, ESR results point towards a doublet ground state16. Later in this section we will account for this and similar observations in terms of competition between JT effects and

5-2 interelectronic repulsions in the anions. Electronic structure calculations of the K8(C60)3

3- cluster have also approximately reproduced the essential features of the C60 NIR/vis absorption spectrum17, though calculational details were not presented, and it is possible that the cause of the manifold of NIR/vis peaks was due to the crystal field within the cluster. (It is for this reason that we avoid the discussion of solid state results in this section.)

15 4- Lawson et al. proposed two alternatives to account for the C60 splitting pattern, one corresponding to a diamagnetic configuration, the other to a paramagnetic triplet electronic configuration.

5- Our own results pertaining to the C60 anion in liquid ammonia appear to be the only results ever obtained for this species18. The absorption spectrum of thin films of

19 6- K6C60 is essentially similar to that of C60 observed here in solution for the first time.

5.2 Experimental Results

2- 3- 4- 5- 6- Figure 5.2 shows the spectra of the C60, C60, C60, C60 and C60 anions obtained in liquid ammonia using the titration technique described in § 3.1.3. As explained there, it was

1- not possible to obtain a spectrum of C60 in this solvent. That ion pairing in solution is not significantly influencing the spectra is suggested by the fact that the spectra were identical regardless of the alkali metal used; solutions were at any rate quite dilute,

-3 6- being of the order of 10 mol/L. The C60 spectrum shows a broad and weak peak at ~6500 cm-1 due to a low concentration of solvated electrons. The identity of the lower anions is confirmed by comparison with spectra in other solvents, as well as by the observation of isosbestic points (see § 3.1.3) which indicate that only two absorbing species were in equilibrium in solution at any one time.

5-3 - e (NH 3) 6- C60

5- C60

4- C60

Proportional Absorbance Proportional 3- C60

2- C60

Solvent

5000 10000 15000 20000 25000 30000 -1 Wavenumber (cm )

n- Figure 5.2: Spectra of C60 (n = 2 - 6) obtained in liquid ammonia at 210 K using sodium metal as the reductant. The illustrated solvent component has been subtracted from each spectrum. The broad peak at 6500 cm-1 in the 6- C60 spectrum is due to a low concentration of solvated electrons.

5-4 Figure 5.2 may be compared with the compilation of C60 anion spectra reported by

20 2- 3- Baumgarten et al. . For the C60 and C60 anions there is clearly some degree of correlation with our spectra, though the spectra do not agree well in detail; peak energies and the lack of multiple NIR/vis peaks suggest that the spectrum assigned by those

4- 6- workers to the C60 anion is more likely to be due to the C60 anion, while the spectrum

6- they assign to the C60 anion is difficult to ascribe. However, our spectra are similar to

2- 3- 4- 13 15 C60, C60 and C60 spectra observed in spectroelectrochemical studies , and the controlled, reversible nature of the titration and the observation of isosbestic points (see § 3.1.3) give good reason to place confidence in the results presented here.

Comparison of Figure 5.1 and Figure 5.2 allows assignment, at least in a broad sense, of the observed absorption peaks. There can be little doubt that all the low energy -1 transitions in the range 7000 - 16000 cm (NIR/vis) are associated with the t1u → t1g transition, and they will be referred to as such elsewhere in this work. These, and the weak features from 16000 to 22000 cm-1 (visible region) to do with the symmetry forbidden hu → t1u transition becoming allowed upon admixture of vibrational components, are in accord with the calculations and assignment of Heath et al.13. The strong peaks above 22000 cm-1 (near ultraviolet) are rather interesting. In each spectrum, this region appears to involve essentially two peaks, both of which shift to lower energy as the anion charge increases. The intensity of the higher energy peak decreases at the expense of the lower energy peak in the higher anions, and is absent in 6- → C60. If we assign the lower energy peak to the (allowed) t1u hg transition, and the higher energy peak to the (allowed) gg/hg → t1u transition, the filling of the t1u levels then accounts beautifully for the change in the relative intensity of the two peaks. Such an assignment leads to the approximate spacing of energy levels proposed in Figure 5.1 -1 (the allowed hu → t1g transition, which would then appear at around 28000 cm , has been calculated to have negligible oscillator strength13).

There are some differences between the current assignment of the UV bands and that of Heath et al.13, as may be seen by a comparison of the orbital spacings in Figure 5.1; in general the spacings inferred by the latter authors are rather smaller. The validity of the present assignment draws much of its strength from the intensity changes that may be seen in the peaks above 22000 cm-1. However, further support comes from the

5-5 21 1- calculations of Chang et al. for the C60 anion, which suggest that the spacing between -1 t1u and t1g levels is ~13500 cm , and more importantly, that the spacing between t1g and -1 upper hg levels is ~11600 cm . Finally, although it is known that the valence orbitals described here are significantly dispersed in the solid (see § 2.2), the overall spacing of bands observed in photoemission (PES) and inverse photoemission (IPES) fulleride studies22 endorse the more widely spaced orbital assignment. It should be noted that the calculations of Heath et al. as applied to the UV region are subject to configuration interaction, and that while their calculations pertain to particular anions, our -1 assignments in Figure 5.1 are more general (for example the stated 11000 cm t1u → t1g gap in Figure 5.1 reflects transitions that occur in the 11000 ± 4000 cm-1 region in Figure 5.2).

2- 6- 1- The similarity of the NIR/vis peaks of the C60 and C60 anions (as well as the C60 anion,

6- see below), and the fact that the C60 anion is incapable of JT distortion, indicate that these three NIR/vis spectra involve only one electronic transition, with an associated

3- 4- 5- vibronic manifold. If this assessment is correct, the C60, C60 and C60 NIR/vis spectra clearly involve multiple electronic origin peaks derived from the t1u → t1g transition, each of which appears to have an associated vibronic manifold.

The general shape of the vibronic manifolds, with a sharp rise in absorbance followed by gradual tailing off of intensity at higher energies, is the kind of Franck-Condon envelope expected for electronic transitions to excited states with similar geometries23, and appears to be generally conserved in all the NIR/vis peaks in all the anions. The misinterpretation of this structure has led to the probably erroneous assignment of electronic transitions in some instances15 24. In the interests of avoiding such confusion, we will therefore describe the vibronic structure first. Such structure, if sufficiently well resolved, is of considerable interest as a potential means of examining the molecular dynamic properties of C60 anions in the absence of a crystal field - an important topic in the context of fulleride superconductivity. For such purposes adequately well resolved vibronic spectra were not obtained, but it was nevertheless possible to considerably

1- 2- enhance the C60 and C60 spectra, as described below.

5-6 One factor that considerably broadens spectra such as are shown in Figure 5.2 is temperature. This operates in a number of ways - higher temperatures allow greater rotational freedom, broadening by Doppler effects, higher energy solvent coordination spheres (the solvent coordination affects the energies of ground and excited states through interaction with the valence orbitals) and thermal population of low energy electronic and vibrational states (causing “hot bands”). For these reasons it was desirable to reduce the temperature as much as possible. Ideally spectra should be measured in the absence of a coordination environment, ie. as a very cold gas, though this is difficult to realize in practice, and at any rate is

2- 25 probably not possible for anions higher than C60 . It is sometimes possible to use a very well defined coordination environment such as in a crystal, or an inert matrix such as solid argon, but it was felt that finding a suitable crystalline

substance into which C60 anions could be doped would be problematic owing to the large molecular volume and the very low reduction potentials involved. Ultimately the best compromise was merely to use a solvent that could be quenched to form a transparent glass.

Figure 5.3: Spectroelectrochemical cell used for the 1- 2- 3- observation of C60 , C60 and C60 anions in 2- methyltetrahydrofuran. A description of the assembly and use is given in the text.

Experiments with liquid ammonia failed to produce clear glasses below the freezing point, and various additives and other glassing solvents and mixtures26 were tried before settling on 2-methyltetrahydrofuran (MTHF) as the most suitable glassing solvent. Because of the difficulties that would arise from the use of this solvent in the liquid ammonia titrator (in particular the insolubility of alkali metals), it was necessary to perform the reduction in other ways. Electrochemical reduction proved to be the most convenient. Spectroelectrochemical reduction of C60 has been performed previously by the author and colleagues to very good effect27, but the fragility of the usual Optically

5-7 Transparent Thin-Layer Electrosynthesis (OTTLE) apparatus limited the temperature at which spectra could be recorded. The type of cell pictured in Figure 5.3 could on the other hand be easily made as required, and was sufficiently robust to withstand several liquid nitrogen quench/thaw cycles. While it was not possible to incorporate a reference electrode in the design, this was not a limitation, since the progress of the reduction (performed at room temperature) could, of course, be monitored spectroscopically. Diffusion between the anode (A) and cathode (B) chambers was clearly undesirable; hence the constriction (C) between the two.

To use the apparatus in Figure 5.3, a microspatula-tip of C60 (not weighed, but less than 1 mg) was admitted to the cathode compartment (cuvette) through the open tap. A small quantity (1-5 mg) of tetrabutylammonium fluoroborate electrolyte (TBABF4) was added in the same way. At this point the apparatus was connected to a vacuum line and pumped to ~10-4 τ using a diffusion pump. The MTHF, dried by standing over sodium wire for several days, was then degassed by boiling under vacuum before being vacuum distilled into the bottom part of the cell, until it just covered the spiral of the anode wire. With the part of the apparatus containing the MTHF immersed in liquid nitrogen and still under dynamic vacuum, the tube was sealed off around the platinum anode wire at D, which had been flattened in this region to help form a better glass to metal seal. In order to be sufficiently long to be put in a flow-tube, it was necessary to break the apparatus at E and attach an additional length of glass rod. Finally, blowing a small hole in the glass just above the anode-to-glass seal at D allowed electrical connection to be made to the anode wire.

After ultrasonication and repeated tipping to dissolve the electrolyte (C60 is practically insoluble in MTHF), an attempt was made to measure a cyclic voltammogram of the cell, using the anode wire as the reference electrode. At voltages up to about 5 V the current was quite negligible, however (only a few nanoamps), and showed no detectable steps in the cyclic voltammogram. The very poor conductivity was disappointing in terms of electrolysis, being indicative of ion-pairing in the electrolyte. Nevertheless, by applying a potential difference of 30 V across the electrodes at room temperature a current of 3 µA could be made to flow. While small, this current was sufficient to produce a distinct reddish colouration in the vicinity of the cathode in the course of a

5-8 1- few hours, and measurement of the spectrum proved this to be due to C60 . Subsequent spectra were measured at intervals of several hours over a period of about a week. It

3- was not possible to reduce further than the C60 anion in the MTHF solution, even the

2- latter anion fairly rapidly decaying back to C60 at room temperature once the potential had been removed.

The anticipated strong effect of temperature on the spectra is shown in Figure 5.4, where

1- 2- a solution containing both C60 and C60 is cooled from room temperature to 77 K. The enhancement of the detail of the spectra upon cooling is clearly evident.

Figure 5.4: Effects of cooling upon spectral resolution in a solution 1- 2- containing the C60 and C60 anions in 2-methyltetrahydrofuran. Arrows indicate the direction of changes on cooling from room temperature to 77 K.

Spectra obtained in this way were generally a mixture of two anions. (Because the 3 potentials of the reduction couples in C60 are fairly evenly separated by ~0.46 V , it follows from the Nernst equation that at equilibrium it is not possible to have a significant quantity of more than two different anions in solution at the same time.) Therefore, after taking account of the solvent, spectra of individual anions were obtained by taking linear combinations of spectra which contained different

5-9 concentrations of two anions. The spectra shown in Figure 5.5 were obtained in this

2- 3- way. The authenticity of the C60 and C60 anions may be judged by comparison with

1- Figure 5.2 as well as references [13] and [15], while comparable C60 anion spectra appear throughout the literature (see for example the compilation by Kondo et al.12).

3- C60

2- C60 Proportional Absorbance 1- C60

Solvent

6000 8000 10000 12000 14000

-1 Wavenumber (cm )

1- 2- 3- Figure 5.5: Near-infrared absorption spectra of the C60, C60 and C 60 anions at 77 K in 2-methyltetrahydrofuran.

Apart from the obvious improvement in the resolution of the spectra that occurs upon

1- 2- 3- cooling (which appears to be considerably more dramatic for C60 and C60 than C60), perhaps the only significant temperature dependent changes can be seen at 9000 cm-1 in the series of spectra shown in Figure 5.4, slightly below the main electronic origin peak → 1- -1 of the t1u t1g transition in C60. A small peak was identified in this position (247 cm below the intense peak) by Bolskar et al.28, who suggested that it may be due to thermal population of a low-lying electronic state, or possibly the lowest vibrational quantum -1 (the Hg(1) mode at 275 cm , see § 6.2), though the available data cannot resolve these two possibilities. As shown in Figure 5.4, in contrast to most of the rest of the

5-10 spectrum, the relative intensity of the absorption at 9000 cm-1 increases with temperature. In terms of possibly thermally populated "ground" states in the present work, it should be remembered that at the temperatures of the ammonia spectra (~210 K), relative thermal populations of upper states drop by a factor of 10 for every 340 cm-1 energy interval, while for the glass matrix work (77 K) the corresponding interval is 120 cm-1. At 4 K the same interval is only 6.4 cm-1, which accounts for the absence of this peak in high resolution inert gas matrix spectra8 9.

5.3 Vibronic Structure

n- Figure 5.6 shows a compilation of the best resolved C60 (n = 1 - 6) anion spectra in the 7000 - 16000 cm-1 region, in which the intense low energy peaks have been shifted to 0 cm-1 in order to allow a visual comparison of the vibronic structure that emanates from the lowest energy electronic origin peaks.

2- Differences between the C60 anion NIR/vis absorption spectra in ammonia (Figure 5.2) and MTHF (Figure 5.5) at the low energy end of the absorption (ie. ~10000 cm-1) are difficult to account for. It should be noted that some details of the fine structure seen in

2- the MTHF C60 anion spectrum, particularly on the low energy side of the most intense

1- peak, are likely to be an artefact of the procedure used to subtract the C60 spectrum. In

2- 13 15 general terms the C60 spectra recorded in dichloromethane and benzonitrile appear

2- more closely similar to the spectrum in ammonia. That the C60 anion's electronic ground state is sensitive to the solvent environment is supported by the work of Boyd et al.14, and we therefore propose that the low energy features may be electronic hot bands arising from the stabilization of different electronic ground states (discussed later) in the (77 K) MTHF matrix.

5-11 6- C60

5- C60

4- C60

3- C60 Proportional Absorbance

2- C60

1- C60

0 2000 4000 6000 8000 Shift (cm-1)

n- Figure 5.6: Superimposed near-infrared spectra of the C60 (n = 1 - 6) t1u → t1g transition manifold, showing modest resolution of vibronic peaks. Traces have been shifted to place the strong lowest energy electronic origin -1 1- 2- peak at 0 cm . The C60 and C60 spectra were recorded in 2-methyltetrahydrofuran at 77 K; the others were recorded in liquid ammonia at ~210 K.

5-12 3- There are differences in the C60 spectra in Figure 5.2 and Figure 5.5 also, these being in the appearance of a possible electronic absorption between the more prominent peaks at 10050 and 12830 cm-1 in Figure 5.5. (The corresponding feature in the ammonia spectrum is perhaps partly due to poorly resolved vibronic structure associated with an electronic transition whose origin is at ~10140 cm-1; the relevant part of the spectrum can be seen more clearly in Figure 5.6). Again, the feature in question appears to be somewhat solvent dependent, being clearly present in dichloromethane13 and rather less so in benzonitrile15.

Because of the small changes in the vibrational modes of C60 upon anion formation (see § 6.2), it is clear that at the presently available resolution, the vibrational structure may be safely treated within the icosahedral (Ih) point group; ie. there is no need to take into account the relatively small molecular distortions that may arise as a result of JT effects, be they static or dynamic. In what follows, the more questionable assumption is therefore that the electronic structure may be approximated within the Ih point group. Fortunately, as we shall shortly see, the new symmetries that could result from JT distortions should not affect which vibrational transitions are allowed.

For the present purposes we are interested in knowing which vibrations can couple to the electronic transitions in the C60 anions. Taking vibronic interactions into account, the electronic transition integral is29: =℘òΨΦ ΨΦ τ pdgg ee , (5.1) where Ψg and Ψe are the ground and excited electronic wavefunctions (t1u and t1g symmetry respectively in Ih), Φg and Φe are the ground and excited vibrational states (Ag and unknown symmetries respectively), and ℘ is the electronic transition operator, which in the Ih point group has t1u symmetry. By elementary symmetry arguments we then find that within the Ih point group, any Φe with Ag, T1g, T2g, Gg or Hg symmetry (in other words, any centrosymmetric vibration) leads to a fully symmetric component in the transition integral, and can therefore participate in vibronic coupling (this is an instance of the Laporte selection rule). A distortion of the molecule in a non- centrosymmetric fashion will lead to other modes becoming vibronically allowed.

However, since the JT distortions are anticipated to occur along the centrosymmetric Hg displacement coordinates (see below), they should not affect the Laporte selection rule.

5-13 This analysis, while simplifying the situation, still leads to 23 allowed centrosymmetric vibrations, which lie in the range 200 - 1600 cm-1 (see § 6.2). Furthermore it is clear from Figure 5.6 that the low intensity structure on the high energy side of the most

1- 2- -1 intense peaks in the C60 and C60 spectra extends well beyond 1600 cm . This fact strongly suggests that combination and overtone bands are necessary to account for the intensity at the higher energies. A possible (and not mutually exclusive) alternative is that a second, electronically forbidden transition lends intensity to the higher energy

1- 2- region of the C60 and C60 transitions shown in Figure 5.6 by becoming allowed through vibronic coupling. Although it is not possible to distinguish these possibilities on the

1- basis of available data, the vibronic analysis of a much better resolved C60 spectrum recorded in a neon matrix has been attempted by Fulara et al.9, and for a more comprehensive assignment the reader is referred to that work (though in light of the vibronic transition symmetry analysis given above, it is felt that modes other than just the ag and hg modes are necessary in its interpretation). It appears from that work that while combination and overtone bands could account for the higher energy vibronic transitions, the assignment of distinct vibronic progressions (such as was attempted in

11 1- the early work of Kato et al. ) is inappropriate. The C60 spectrum in Figure 5.6 appears

+ to be identical to the neon matrix spectrum (which is contaminated with C60) when differences in resolution and the appearance of a weak "hot band" at ~9000 cm-1 in our data are taken into account.

5.4 Near-Infrared Electronic Transitions

We turn now to a discussion of the electronic origins that may be seen in the NIR/vis spectra. Their energies are given for a number of solvents in Table 5.1. It can be seen that the solvent has some effect on the transition energies, with more polar solvents generally exerting hypsochromic shifts.

5-14 -1 Table 5.1: Energies of t1u → t1g Electronic Transitions in C60 Anions (energies in cm )

Anion Solvent 1st Origin 2nd Origin 3rd Origin 4th Origin 1- 28 C60 (various) 9196 - 9310 ------MTHF glass 9305 ------dichloromethane13 10500 ------2- 15 C60 benzonitrile 10505 ------MTHF glass 10520 ------ammonia 10610 ------dichloromethane13 7400 10100 11400 12900 3- 15 C60 benzonitrile 7315 10363 11390 12690 MTHF glass 7345 10050 11620 12830 ammonia 7360 10140 11560 13000 4- 15 C60 benzonitrile 8270 (unresolved?) 13740 -- ammonia 8480 11210 (?) 13940 -- 5- C60 ammonia 9860 13280 -- -- 6- C60 ammonia 10740 ------

n- Lowering of the t1u orbital degeneracy by JT effects is anticipated in those C60 anions with incompletely filled orbitals, and attempts have been made in the literature to describe the manifold of electronic origins in the NIR/vis spectra as arising directly from this phenomenon15. Before attempting a theoretical description of the spectra, however, some account needs to be given to the essential features of the JT effect in the context of the C60 anions.

The partial occupation of the t1u orbitals may lead to a symmetry-lowering JT distortion of the molecule along one or more of its vibrational normal coordinates, such that the valence orbital degeneracy is lifted, and the structure of lowest energy no longer has the full icosahedral symmetry - indeed there are generally several such minima, closely spaced in energy, and corresponding to different distortions of the original molecule.

Molecular distortions that can lead to splitting of the t1u orbitals in C60 may be found by 30 evaluating the direct product within the icosahedral (Ih) point group :

t1u ⊗ t1u = Ag + T1g + Hg 31 (this direct product is incidentally the same for the t1g orbitals ). Ag modes cannot lift the degeneracy, however, and the T1g mode is asymmetric. Thus it is the Hg modes that are held responsible for JT instabilities in C60 anions. The occupancy of the new non- degenerate electronic sublevels will depend on interelectronic interactions on the molecule as well as their energy separation. In actual experiments there is the additional

5-15 complication that external perturbations such as a solvent coordination environment and/or crystal fields are capable of further lowering the degeneracy of the valence orbitals.

Distortion of the icosahedral C60 molecule along individual Hg mode coordinates leads 35 to structures with D5d, D3d and D2h symmetries (though in general the distortions may occur simultaneously along several such coordinates, leading to still lower molecular

1- symmetries). For the C60 anion, the D5d, D3d and D2h structures are essentially degenerate, each being about 8.4 kJ/mol more stable than the undistorted ion35. The magnitude of the geometrical distortions of the molecules is calculated by Green et al.25 to be sufficiently small that the molecule can vibrate through icosahedral symmetry even at the zero point level, ie. the JT distortions are expected to be dynamic32 33 34 (for the

1- 35 C60 anion, the relevant bond length changes are calculated to be ~1% ).

There is still significant disagreement about the magnitude of the effect from an electronic standpoint. At the time of writing it appears that there have been only two 25 35 serious attempts to calculate the JT effect in C60 anions , reference [35] dealing

1- -1 solely with the C60 anion and predicting a split of ~2200 cm in the t1u-derived sublevels. Splittings of this order of magnitude are quite comparable to those seen in the 7000 - 15000 cm-1 region of our spectra. On the other hand, the more computationally intense calculations in reference [25] indicate that the t1u levels are split

-1 n- by at most ~800 cm in any of the C60 (n = 1 - 6) anions (their calculated orbital energy diagrams are reproduced in Figure 5.7), an amount which is too small to be directly useful in the interpretation of the spectra presented here, at least in terms of the analysis of Lawson et al.15.

5-16 Figure 5.7: Splitting patterns of the hu, t1u and t1g orbitals that occur in the n- ≡ -1 C60 (n = 1 - 6) anions as a result of symmetry lowering. (1 V 8066 cm ). 4- For the C60 anion, calculations were only applied to the illustrated triplet state; experimental data indicates the ground state is a spin singlet. (From Green et al.25)

The illustrated magnitude of energy splitting is of interest in that it lies in the -1 intramolecular vibrational energy range of the C60 molecule (roughly 200 - 1600 cm , see § 6.2), a fact which can potentially lead to strong interactions between electronic and vibrational transitions (ie. a breakdown of the Born-Oppenheimer approximation)36. It is then possible that electron-phonon interactions contribute significantly to the electronic structure of the fullerene anions37. The analysis of the electron-phonon interaction is a matter of considerable complexity, however38, and a satisfactory reconciliation of experimental and theoretical results seems unlikely at the present time. In what follows we merely attempt a possible explanation of the observed spectra in terms of competition between interelectronic repulsions and the JT effect.

n- Table 5.2 shows the multiplicity of electronic states in C60 (n = 1 - 6) anions that result from interelectronic repulsion for icosahedral geometry. States arising from a particular configuration of t1u electrons have been given the same colour; note the

5-17 2 4 1 5 complementarity of the t1u and t1u configurations, and also of the t1u and t1u

39 3 1 1 2 4 configurations . The T1g, Hg and Ag states for the t1u and t1u configurations in Ih symmetry correspond directly to the 3P, 1D and 1S states of a p2 or p4 system (spherical

4 2 2 3 symmetry), while the Au, Hu and T1u states for the t1u configuration correspond to the 4S, 2D, and 2P states of a p3 system. (Here the states are written from lowest to highest energy according to Hund's rule.)

Table 5.2: Possible spin configurations in C60 anions Anion Ground States Excited States 1- 1 2 1 2 C60 (t1u ) T1u (t1g ) T1g 2- 2 3 1 1 1 2 ⊗ 1 2 C60 (t1u ) T1g + Hg + Ag (t1u ) T1u (t1g ) T1g 3- 3 4 2 2 2 3 1 1 ⊗ 1 2 C60 (t1u ) Au + Hu + T1u (t1u ) T1g + Hg + Ag (t1g ) T1g 4- 4 3 1 1 3 4 2 2 ⊗ 1 2 C60 (t1u ) T1g + Hg + Ag (t1u ) Au + Hu + T1u (t1g ) T1g 5- 5 2 4 3 1 1 ⊗ 1 2 C60 (t1u ) T1u (t1u ) T1g + Hg + Ag (t1g ) T1g 6- 6 1 5 2 ⊗ 1 2 C60 (t1u ) Ag (t1u ) T1u (t1g ) T1g

Before describing the possible electronic transitions that may be occurring, we will briefly review the electronic ground state configurations as determined by electron spin resonance (ESR) spectroscopy and magnetic susceptibility results. There can be no

1- 5- question that the C60 and C60 anions have doublet ground states. The work of

14 2- Boyd et al. indicates that C60 has three electronic "ground" states whose ESR spectra correspond to a doublet-like resonance and two triplets, one of the triplets representing a

2- low-lying thermally populated state (note that C60 has no doublet state). Magnetic

2- susceptibility studies of C60 salts (in which crystal field splitting appears to influence the ordering of multiplet states14) indicate a singlet ground state, but that the triplet state is

16 3- significantly thermally populated even at 6 K . C60 appears to have a doublet ground state both in solution16 20 and in the magnetically "dilute" salt [Na(crown) 16 (THF)2]3[C60] ; in solution there is evidence for a higher energy but thermally accessible doublet ground state separated by less than 1 kcal/mol (350 cm-1) from the

16 4- 20 true ground state . The C60 anion appears to be diamagnetic both in solution and in 40 the salt Rb4C60 (in the latter, the thermally activated paramagnetism involves a barrier

5-18 of ~830 (±70) cm-1), and thus appears to have a singlet ground state. Table 5.3 summarizes these observations.

Table 5.3: Observed ground state multiplet structures in C60 anions (see text) Anion Ground Spin Multiplet Comment 1- C60 doublet 2- C60 singlet + two triplets there is a third, higher energy but thermally (practically degenerate) accessible triplet state 3- C60 doublet there may be another thermally accessible doublet 4- C60 singlet 5- C60 doublet 6- C60 singlet

A growing number of ground state multiplet structure calculations have been performed for the higher anions25 41 42. Of these, reference [41] does not account well for the anion ground states as just described, and there are very large discrepancies between calculated and observed transition energies, but it nevertheless serves to illustrate the complexity of the situation even before JT effects have been considered. Reference [42] calculates the possible multiplet structures that may arise from different electron configurations in the t1u orbitals, taking into account the configuration interaction with adjacent electronic states. The near degeneracy of singlet and triplet ground states in

2- 3- C60 is correctly predicted, but the ground state of C60 is calculated to be a quartet, with a large gap (1900 cm-1) between the quartet state and the lowest calculated doublet state.

Again, the symmetry is constrained to the Ih point group, so that JT effects are not considered. Finally, the recent calculations of Green et al.25 (Figure 5.7), in which the anions were allowed unconstrained geometrical distortion, generally appear to be consistent with the observed ground state spin multiplets, though the calculations did not include all possible multiplet structures and the work gives little detail about their relative energies.

3 1 2 2 Returning now to Table 5.2, the electronic microstates of the T1g, Hg, Hu and T1u states will cease to be degenerate once the molecular symmetry has been lowered from icosahedral by JT distortions. The manner in which the spin states are split for an axial

5-19 2 3 distortion is shown in Figure 5.8 for the t1u system and in Figure 5.9 for the t1u system. Here K is the one-centre exchange integral,

K= t1ux()12 t 1uy ( ) t 1uy () 12 t 1ux ( ), (5.2) and for a pure atomic p orbital K = 3F2, where F2 is the interelectron repulsion parameter of Condon and Shortley43. The parameter v represents the static ligand field 2 2 2 produced by an axial distortion of Hg(2z -x - y ) symmetry. (Note the simplification that the distortion only occurs along one vibrational coordinate; as already stated, the JT distortion may involve simultaneous distortion along several coordinates. It appears, 25 however, that the Hg(2) mode plays the dominant role in the distortion .)

It is clear that depending on the relative magnitude of the JT distortion and the interelectronic repulsion, Figure 5.8 and Figure 5.9 can be used to account for the experimentally observed multiplet ground states as summarized in Table 5.3.

In order to attempt a description of the NIR/vis spectra, we will make the assumption that multiplet structure in the excited states of the anions is determined predominantly

n-1 by interelectronic repulsions within the t1u configuration, and that the coupling between

n-1 1 states derived from the t1u and t1g configurations is much smaller. Thus, for example,

3- we assume that the ordering of electronic excited states of the C60 anion (see Table 5.2)

2 will be given by the interactions relevant to the t1u system shown in Figure 5.8 and

5- should be similar to that of the C60 anion. However, the distribution of the electric dipole intensity will be dictated by the number and the equilibrium geometry of the

3- 5- ground levels, and these are obviously very different for C60 and C60.

5-20 1A K

8

6 1A g 1E

4 3E 1 Hg 2

3 T1g v/K

-10-8-6-4-2 2 4 6 810

1 -2 A

3E -4 1 A 1E

-6 3A 3 Figure 5.8: Changes in the energy of the three electronic spin states ( T1g, 1 1 2 Hg and Ag in Ih symmetry) of the t1u system upon lowering the symmetry by an axial Jahn-Teller distortion. Positive and negative v correspond to different phases of the distortion.

K 2E 10

8

2 6 T1u 2A 4 2 Hu 2E, 2A 2

4 Au 0 4A

-2

2E -4 0246|v|/K 4 Figure 5.9: Changes in the energy of the three electronic spin states ( Au, 2 2 3 Hu and T1u in Ih symmetry) of the t1u system upon lowering the symmetry by an axial distortion.

5-21 n-1 1 2 A given state with spin S' in the t1u configuration will couple with (t1g ) T1g to give states with resultant S = S' ± 1/2. Hence transitions to states derived from all the parent

n-1 states of the t1u configuration are usually spin-allowed. The only exceptions are: 3- 3 4 4 → 3 ⊗ 2 a) for C60, transitions from (t1u ) A2u are restricted to only A2u T1g T1g, and 4- 4 ⊗ 2 b) for C60, transitions from ground spin singlets to A2u T1g are forbidden.

1- 6- The C60 and C60 anions have what appears to be a single-origin NIR absorption and

1- relatively simple electronic structures, and so we will treat them first. In the C60 anion there are no interelectronic repulsions in the ground or excited states, since the t1u and t1g orbitals contain at most one unpaired electron. The single electronic transition that is 2 observed does not tell us whether JT distortions are occurring or not; if the T1u ground state splits owing to symmetry lowering to D5d or D3d (see Figure 5.10), and if only the lower of the two ground states is populated, then for either kind of distortion only the 12 transition A2u → E(1)g is allowed .

Figure 5.10: Splittings of T1u and T1g-derived states upon lowering the symmetry from Ih to D5h or D3d. Arrows show the allowed transitions from the ground state.

Estimates of the splittings of the two ground states vary enormously, from ~70 cm-1 (reference [11]) to ~2200 cm-1 (reference [35]), and there is clearly little consensus in the literature as to the magnitude of this JT splitting. If the upper ground state is thermally populated, we can anticipate "hot bands" in the spectra. As already discussed, there does appear to be a very weak hot band on the low energy edge of the well

1- resolved 77 K C60 NIR absorbance, but the energy difference between it and the adjacent very intense peak is such as to make it look suspiciously like a hot band arising from the thermal population of the lowest energy vibrational state (the Hg(1) mode).

5-22 6- 2 ⊗ 2 1,3 1,3 1,3 For the C60 anion, the excited state multiplet is T1u T1g = Au + T1u + Hu. Only 1 1 1 one transition from the A1g ground state is electric dipole allowed, A1g → T1u. The

1- sideband structure is probably vibronic as in C60, but may also reflect small distortions 1 of the JT-active T1u state.

2- Turning to the C60 spectrum, we see that to a first approximation the NIR/vis absorption

1- 6- peak profile resembles that in the C60 and C60 spectra. As far as the ground multiplet is concerned, Figure 5.8 suggests that a JT distortion (v/K ≈ -7) could lead to the coexistence of a triplet (3E) and a singlet (1A) state; distortion to symmetry lower than 3 3 D5d or D3d may split the E state into two closely spaced A states, consistent with the multiplet ground state data summarized in Table 5.3. The existence of another thermally populated triplet state is also possible if one recalls that the distortion may involve positive or negative v.

Our basic assumption of weak interactions between electrons in t1u and t1g orbitals 2- 2 ⊗ 2 1,3 1,3 1,3 implies that the splitting of C60's excited states T1u T1g = Au + T1u + Hu should be small. Since the ground levels S = 1 and S = 0 are closely spaced, we expect the

2- absorption profile for C60 to have no large splitting, in agreement with the basic absorption profile. The various thermally accessible "ground" multiplet states, as well as thermal population of the lowest vibrational quantum, nevertheless lead to the

2- possibility of hot bands in the C60 NIR/vis spectrum, and there is strong evidence for such bands, particularly in the 77 K spectrum.

5- 2 6- The C60 anion has a T1u ground state, in which (as already described for the C60 anion) the JT splitting is believed to be small, resulting in a 2A ground level25. The excited

4 1 states based on the t1u t1g configuration are now expected to display some electronic fine

4 structure because the parent states in the t1u configuration have significant splitting coming from interelectronic repulsion (see Figure 5.8). Since the ground level is supposed to be 2A (corresponding to positive v), a rough distribution of the Franck- Condon maxima in the absorption spectrum may be represented by a vertical section in Figure 5.8 for v/K > 0. Each of the states in Figure 5.8 should be considered to be split

4 1 still further, owing to the coupling of t1u with t1g in the point group corresponding to the JT distorted state, though such splitting may not be large. In principle the NIR/vis

5-23 spectrum could therefore be very complex; the observation of only two clear features in the measured spectrum perhaps indicates that many of the transitions are associated with small oscillator strengths.

4- As indicated in Table 5.3, the C60 anion is believed to have a singlet ground state, so that from Figure 5.8 we anticipate a large and negative value of v/K. The excited states

3 1 based on the t1u t1g configuration are expected to have moderately large splitting coming

3 from the t1u configuration (Figure 5.9). Since the ground state has a large distortion, the Franck-Condon maxima in the absorption spectrum should correspond to a vertical section in Figure 5.9 at a large value of |v|/K. Again, each of the doublet states (2Γ) in

1 Figure 5.9 should be interpreted to be split further owing to coupling to the JT-active t1g configuration, which will lead to allowed spin singlets. Figure 5.9 suggests that there are two widely spaced groups of allowed spin singlets, as well as several allowed states of intermediate energy, in rough agreement with the observed spectrum.

3- The C60 ion has a doublet ground state that requires a large JT distortion with |v|/K > 4 according to Figure 5.9. The excited states are expected to have significant splitting

5- 3- from interelectron repulsions, just as for the C60 anion. For C60 the observed Franck- Condon maxima will correspond to vertical sections in Figure 5.8 for both positive and negative values of v/K, so that a rich electronic fine structure can be anticipated.

A more meaningful analysis of this model requires calculations of dipole strengths for all the electronic transitions. Furthermore, in order to eliminate contributions from electronic hot bands unambiguously, measurements at liquid helium temperatures are essential. The combination of interelectronic repulsions and JT distortions can nevertheless lead to a plausible qualitative account of the complexity of electronic

n- transitions in the C60 (n = 1 - 6) anions' NIR/vis spectra.

5-24 5.5 References

1 P.D. Hale, J. Am. Chem. Soc., 108, 6086 (1986) 2 R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Lett., 125, 459 (1986) 3 Q. Xie, E. Pérez-Cordero and L. Echegoyen, J. Am. Chem. Soc., 114, 3978 (1992) 4 Y. Ohsawa and T. Saji, Chem. Commun., 781 (1992) 5 M.A. Greaney and S.M. Gorun, J. Phys. Chem., 95, 1742 (1991) 6 J. Stinchcombe, A. Pénicaud, P. Bhyrappa, P.D.W. Boyd and C.A. Reed, J. Am. Chem. Soc., 115, 5212 (1993) 7 O. Gunnarsson, H. Handschuh, P.S. Bechthold, B. Kessler, G. Ganteför and W. Eberhardt, Phys. Rev. Lett, 74, 1875 (1995) 8 Z. Gasyna, L. Andrews and P.N. Schatz, J. Phys. Chem., 96, 1525 (1992) 9 J. Fulara, M. Jakobi and J.P. Maier, Chem. Phys. Lett., 211, 227 (1993) 10 T. Kato, T. Kodama, T. Shida, T. Nakagawa, Y. Matsui, S. Suzuki, H. Shiromaru, K. Yamauchi and Y. Achiba, Chem. Phys. Lett., 180, 446 (1991) 11 T. Kato, T. Kodama, M. Oyama, S. Okazaki, T. Shida, T. Nakagawa, Y. Matsui, S. Suzuki, H. Shiromaru, K. Yamauchi and Y. Achiba, Chem. Phys. Lett., 186, 35 (1991) 12 H. Kondo, T. Momose and T. Shida, Chem. Phys. Lett., 237, 111 (1995) 13 G.A. Heath, J.E. McGrady and R.L. Martin, Chem. Commun., 1272 (1992) 14 P.D.W. Boyd, P. Bhyrappa, P. Paul, J. Stinchcombe, R.D. Bolskar, Y. Sun and C.A. Reed, J. Am. Chem. Soc., 117, 2907 (1995) 15 D.R. Lawson, D.L. Feldheim, C.A. Foss, P.K. Dorhout, C.M. Elliott, C.R. Martin and B. Parkinson, J. Electrochem. Soc., 139, L68 (1992) 16 P. Bhyrappa, P. Paul, J. Stinchcombe, P.D.W. Boyd and C.A. Reed, J. Am. Chem. Soc., 115, 11004 (1993) 17 K. Coutinho, S. Canuto, A. Fazzio and R. Mota, Modern Phys. Lett. B, 9, 95 (1995) 18 W.K. Fullagar, I.R. Gentle, G.A. Heath and J.W. White, Chem. Commun., 525 (1993) 19 V.I. Srdanov, A.P. Saab, D. Margolese, E. Poolman, K.C. Khemani, A. Koch, F. Wudl, B. Kirtman and G.D. Stucky, Chem. Phys. Lett., 192, 243 (1992) 20 M. Baumgarten, A. Gügel and L. Gherghel, Adv. Mater., 5, 458 (1993) 21 A.H.H. Chang, W.C. Ermler and R.M. Pitzer, J. Phys. Chem., 95, 9288 (1991) 22 T. Takahashi, S. Suzuki, T. Morikawa, H. Katayama-Yoshida, S. Hasegawa, H. Inokuchi, K. Seki, K. Kikuchi, S. Suzuki, K. Ikemoto and Y. Achiba, Phys. Rev. Lett., 68, 1232 (1991) 23 "Group Theory in Spectroscopy", by S.B. Piepho and P.N. Schatz, ©1983 J. Wiley and Sons, Inc., Brisbane, Australia 24 B. Friedman and R.Q. Luo, Phys. Rev. B, 51, 7916 (1995) 25 W.H. Green Jr., S.M. Gorun, G. Fitzgerald, P.W. Fowler, A. Ceulemans and B.C. Titeca, J. Phys. Chem., 100, 14892 (1996) 26 "Low Temperature Spectroscopy; Optical Properties of Molecules in Matrices, Mixed Crystals and Frozen Solutions", by B. Meyer, ©1971 American Elsevier Publ. Co., N.Y. 27 W.K. Fullagar, "Buckminsterfullerene: C60" (Honours thesis), Australian National University, 1992 28 R.D. Bolskar, S.H. Gallagher, R.S. Armstrong, P.A. Lay and C.A. Reed, Chem. Phys. Lett., 247, 57 (1995) 29 "Orbitals and Symmetry", by D.S. Urch, ©1970 Penguin Books, Ringwood, Victoria, Australia

5-25 30 C.M. Varma, J. Zaanen and K. Raghavachari, Science, 254, 989 (1991) 31 A.R. Kortan, N. Kopylov, S. Glarum, E.M. Gyorgy, A.P. Ramirez, R.M. Fleming, F.A. Thiel and R.C. Haddon, Nature, 355, 529 (1992) 32 B. Gotschy, M. Kiel, H. Klos and I. Rystau, Sol. St. Commun., 92, 935 (1994) 33 M. Bennati, A. Grupp and M. Mehring, J. Chem. Phys., 102, 9457 (1995) 34 X. Wei and Z.V. Vardeny, Phys. Rev. B, 52, R2317 (1995) 35 N. Koga and K. Morokuma, Chem. Phys. Lett., 196, 191 (1992) 36 Y. Asai, Chem. Phys. Lett., 195, 551 (1992) 37 L. Bergomi and Th. Jolicoeur, preprint of "Electronic structure of fullerene anions"; submitted to Phys. Rev. B (1993) 38 N. Manini, E. Tosatti and A. Auerbach, Phys. Rev. B, 49, 13008 (1994) 39 S. Chakravarty, M.P. Gelfand and S. Kivelson, Science, 254, 970 (1991) 40 I. Lukyanchuk, N. Kirova, F. Rachdi, C. Goze, P. Molinie and M. Mehring, Phys. Rev. B, 51, 3978 (1995) 41 F. Negri, G. Orlandi and F. Zerbetto, J. Am. Chem. Soc., 114, 2909 (1992) 42 R. Saito, G. Dresselhaus and M.S. Dresselhaus, Chem. Phys. Lett., 210, 159 (1993) 43 "Theory of Atomic Spectra" 2nd Edn., by E.H. Condon and G.H. Shortley, ©1953, Cambridge University Press

5-26 6. Dynamics

One of the main theories attempting to explain the superconductivity in certain fullerides postulates its origin in the coupling of vibrational and electronic motions of the C60 anions. This mechanism, described in § 2.3.5, requires a detailed understanding of the vibrational properties of the fullerides, which are examined here.

The relatively rigid nature of the fulleride molecular ions means that to a good approximation, the dynamics of the solid materials are of two types: intermolecular dynamics, in which C60 molecules are imagined to be rigid icosahedra, and intramolecular dynamics, in which the vibrations of C60 molecules are treated independently of the motions of the surrounding lattice. In principle, conduction electrons can couple to either kind of motion. The aim of this chapter is to present a number of inelastic neutron scattering measurements, probing both kinds of motions, in the context of the other recent literature on this subject.

6.1 Intermolecular dynamics

There are three popular experimental approaches to the low energy dynamics of the fullerides; Raman spectroscopy, far infrared spectroscopy, and inelastic or quasielastic neutron scattering. Raman and infrared techniques are subject to the usual selection rules and are restricted to the description of Brillouin zone-centre modes.

6.1.1 C60 Intermolecular Dynamics

For the Pa 3 structure of low temperature undoped C60, there are 24 low frequency acoustic and optic modes with the following Brillouin zone-centre symmetries1:

Γvib = Ag + Eg + 3Tg + Au + Eu + 3Tu.

The acoustic modes have Tu symmetry; nine of the other modes are displacive, with symmetries Au + Eu + 2Tu, while the remaining twelve modes are rotational, with

6-1 symmetries Ag + Eg + 3Tg. The Tu modes are infrared active, while all the g-modes are Raman active. The eigenvectors and energies associated with these modes are illustrated in Figure 6.12.

Figure 6.1: Forms of intermolecular normal modes in Pa 3 C60. Arrows show the displacement vectors for translational modes (u-symmetry) and axial rotation vectors for the librational modes (g-symmetry). Zone-centre energies are given in cm-1. (From Lipin and Mavrin2.)

3 Raman studies of single-crystal C60 at 90 K show transitions at energies in quite good agreement with the gerade- modes illustrated in Figure 6.1. Infrared studies of lattice 1 4 5 modes in undoped C60 have been complicated by atmospheric contamination ; thorough degassing leads to the appearance of only the two infrared-active Tu modes, which show a slight softening as the temperature is raised6.

The calculated density of states (DOS)2 for the intermolecular modes is shown in Figure 6.2. Other authors have presented similar results7 8 9. The general structure of the DOS simulation in Figure 6.2 is supported by a number of experimental reports10 11, the inelastic neutron scattering results (200 K) of Renker et al.12 being reproduced in the top panel of Figure 6.3. Differences between theoretical and experimental curves have been accounted for in terms of multiple phonon scattering and/or orientational disorder (brought about by thermal population of the libronic phonons)19 13 14. Low energy 15 undoped C60 inelastic neutron scattering (INS) data have also been collected at 30 K . These data have a relatively sharp peak at 22 cm-1, and agree much better with the calculated DOS shown in Figure 6.2. Temperature dependent neutron16 and Raman17 studies of the librational modes of Pa 3 C60 both clearly show energy broadening and shifting to lower energy of the overall intensity of librational peaks as the temperature is raised. Shifts to higher energy, observed upon application of pressure, have been used

6-2 in conjunction with the data showing shifts to lower energy as the temperature is raised, to conclude that the latter are mainly due to the thermal population of libronic phonons, with a smaller contribution from the shallower potential wells that result from lattice expansion18.

Figure 6.2: Density of intermolecular vibrational states calculated for Pa 3 2 C60 by Lipin and Mavrin . The dashed line represents the density of librational modes, the solid line is for the translational modes.

Pintschovius and Chaplot have measured C60's intermolecular vibrational dispersion curves, which appear to be in reasonable agreement with calculated curves19.

6.1.2 Alkali Fulleride Intermolecular Dynamics

Surprisingly, only three incomplete calculations of the lattice dynamics of the fcc A3C60 fullerides have been presented in the literature20 21 22. Brillouin zone-centre mode frequencies have recently been calculated both for the intermolecular and intramolecular 23 zone-centre modes of K6C60 . The lack of theoretical attention probably reflects the widely held belief that intermolecular modes are not responsible for electron pairing in the superconducting fullerides, as well as difficulties in the preparation and handling of

6-3 large single crystal fulleride samples suitable for experimental studies of vibrational dispersion curves.

When compared to C60, the presence of the alkali metal ions in the fullerides leads to hardening of the librational modes and the appearance of new peaks in the region 70 - 140 cm-1, as illustrated in the lower two panels of Figure 6.3. Raman scattering in the

Figure 6.3: Inelastic neutron scattering from C60, K3C60 and Rb3C60. Closed and open circles show energy loss and gain measurements, respectively. Thin solid lines show the calculated contributions from translational modes (the dashed curve in C60 uses force constants fitted to measured dispersion curves10). Contributions of the alkali atoms (A(1): tetrahedral sites, A(2): octahedral sites) are indicated by hatched areas. The dash-dotted curve is the difference between measured and calculated results, and gives an 12 estimate of the C60 librational modes. (Modified from Renker et al. )

24 12 intermolecular region in AxC60 compounds confirms these results. Renker et al. have fitted their inelastic neutron data using force constants that lead to positioning of

6-4 alkali metal optic phonons in the positions indicated by the shaded areas in Figure 6.3. These may be divided into two groups, depending on whether the metal atoms in the octahedral or tetrahedral sites are primarily involved - the higher M-C force constants for M in a tetrahedral site lead to the higher energy peak in the region 70 - 140 cm-1, while that part of the DOS involving the octahedral M is predicted to be buried under -1 the main C60 band between 0 and 70 cm . The K3C60 lattice dynamics calculations of Belosludov and Shpakov20 are also relevant here; their calculations indicate a DOS due

3- -1 + 3- -1 to C60 librations at ~26 cm , and K and C60 translational motions at ~45 - 130 cm , with a gap from ~70 - 110 cm-1. Above 130 cm-1 is another gap in the DOS until -1 ~260 cm (the lowest energy C60 intramolecular mode, Hg(1)). More recent molecular dynamics calculations on K3C60 have been frustrated by possible anharmonicity in the octahedral K atom's potential well22.

6.1.3 TFXA Samples and Sample Cell Contributions

Three INS data sets were recorded by us on the time of flight crystal analyser spectrometer (TFXA, see § 8.2.2), and are described in conjunction with the data of other workers throughout this chapter. Details of run times, temperatures, sample cans and contaminants are summarized in Table 6.1. Scattering contributions from sample containers fall mostly in the fulleride intermolecular region, and so are described here.

For practical reasons the empty sample cans were not run on the TFXA instrument. Figure 6.4 and Figure 6.5 compare the relevant fulleride data with TFXA spectra of aluminium and vanadium, showing their perceptible but small contributions to our data. Aluminium scatters in both the intermolecular (<200 cm-1) and intramolecular (>200 cm-1) regions, while vanadium scatters mostly in the intermolecular region. Although in principle it is possible to quantitatively subtract such contributions, we find in § 6.2.3 that similar subtractions do not justify the results. Spectra in Figure 6.4 and Figure 6.5 have therefore been scaled so as to allow visual comparison of relative peak intensities, and do not reflect the quantity of material in the beam.

6-5 Table 6.1: Summary of the TFXA samples

Sample Label [Size (g)] Container (Time (µAh)) + {Temp (K)} geometry Notes/contaminants C60 "No impurity was detectable by HPLC, IR, UV or MS. [1.6] The material was heated at 440 K under primary vacuum (6513) ? for 12 hr, stirred with C6D6 (2 mL/g of C60) for 20 min, {25} dried, and heated again under primary vacuum three times for 12 hr."25 Na2C60 0.3 mm Prepared by direct reaction of Na with C60 in an evacuated [2] Al pyrex tube at 340ºC for five days, with intermittent (8770) 55×25×2 shaking to ensure homogeneity. Main impurity is undoped {5} mm plate C60 (10% - 27% by weight), but also contains at least one unknown crystalline impurity phase - see § 4.3.1 Rb3C60 V cyl. Crude soot from the electric arc was used in the synthesis [2.2] 10 mm ø after washing with ether, extraction with toluene, and (tot. 8461) 40-50 mm outgassing at 290ºC. C70 impurity level ~6%. Sharp {see *} deep neutron diffraction patterns indicated that the synthesis "appears to have been at least partially successful"26 Rb4C60 0.3 mm Preparation and crystallographic fits are detailed in § 3.1.1 [3] Al and § 4.3.2, respectively; assumption of stoichiometric (8623) 55×25×2 phases implies 30% Rb3C60, 66% Rb4C60 by weight, the {5} mm plate remainder being an amorphous, Rb-free component. Rb3C60(ND3)2.5 V cyl. Gaseous ammonia doping of Rb3C60 led to product [2] 10 mm ø containing Rb3C60(ND3)4.8 (46 wt.%, poorly crystalline), 27 (7583) 40-50 mm Rb1C60 (29 wt.%) and Rb4C60(ND3)1.7 (25 wt.%) {5} deep Rb6C60 9 mm ø “Rb6C60 was prepared by reaction of C60 with Rb metal in [0.6] silica tube a sealed, evacuated pyrex tube at 250°C (4 days), 300°C (4 (6387) days), followed by distillation of excess Rb away from the {20} product. Phase purity was confirmed by X-ray and neutron diffraction.”28

* Data consists of four runs in the temperature range 22 K to 290 K: (1023.0 µAhr for 290 K to 147 K, 1351.4 µAhr for 112 K to 30 K, 2606 µAhr at 22.4 K and 3214.6 µAhr at 35 K). In the first of these the data is particularly poor and the low energy librational modes are particularly intense. Except where otherwise indicated, illustrated data shows the summation of the three low temperature runs. Possible slight temperature-dependent changes of the high energy intramolecular modes are discussed in § 6.2.7.

6-6 Rb4C60

Na C

Neutron Intensity 2 60

Aluminium

100 150 200 250 300 350 400 Energy Transfer (cm-1)

Figure 6.4: Contribution of aluminium to the Na2C60 and Rb4C60 inelastic neutron spectra. The spectra have been offset for clarity; horizontal lines show the zero counts levels.

Rb3C60

Rb3C60 (ND3) 2.5 Neutron Intensity

Vanadium

50 100 150 200 250 Energy Transfer (cm-1)

Figure 6.5: Contribution of vanadium to the Rb3C60 and Rb3C60(ND3)2.5 inelastic neutron spectra. The spectra have been offset for clarity; horizontal lines show the zero counts levels.

6-7 6.1.4 TFXA Intermolecular Inelastic Neutron Scattering

The TFXA spectra from the intermolecular modes of Na2C60, Rb3C60, Rb4C60 and

Rb3C60(ND3)2.5 are compared in Figure 6.6.

Rb3C60 (ND3) 2.5

Rb4C60

Rb3C60 Neutron Intensity

Na2C60

50 100 150 200 Energy transfer (cm-1)

Figure 6.6: TFXA inelastic neutron scattering spectra of Na2C60, Rb3C60, Rb4C60 and Rb3C60(ND3)2.5 in the intermolecular region.

The TFXA spectrometer resolution is insufficient to give a good account of the C60 librational DOS below ~30 cm-1, and for such low energy INS spectra we refer the reader to the work of Renker et al.12 such as is shown in Figure 6.3. (Other spectra of 12 Renker et al. appear to show some broadening of the librational part of K3C60's DOS -1 between 8 and 24 cm at temperatures below the superconducting Tc, suggesting an involvement of the librations in the superconductivity mechanism. As described in

6-8 § 2.3.5, normal state resistivity measurements also provide some evidence for the coupling of conduction electrons to low energy phonons.) While our TFXA results do not allow a meaningful comparison of the librational DOS in the fullerides, they do give a considerably better picture of the 70 - 180 cm-1 alkali optic phonons.

Between 30 and 70 cm-1, the four spectra shown in Figure 6.6 are quite different. The 15 essential features of the "Na2C60" spectrum are not much changed from undoped C60 , though the similarity may be partly due to the ~25% contamination of C60 in the former. In contrast with the data of Renker et al. (Figure 6.3) which show a pronounced -1 minimum at ~45 cm , the Rb3C60 TFXA data show an essentially featureless rise below -1 70 cm , regardless of the temperature of the data collection. The "Rb4C60" spectrum shows a sharp peak at 40 cm-1, as well as features at about 30 and 55 cm-1, which are absent in the Rb3C60 spectrum and therefore cannot be due to the Rb3C60 impurity. At the present time no lattice dynamics calculations are available for the Rb4C60 structure, but since all its Rb+ ions are in distorted tetrahedral sites, we may postulate on the basis -1 of Figure 6.3 that the features below 60 cm in the "Rb4C60" spectrum have little contribution from alkali optic modes.

The overall similarity of the spectra in the 70 - 180 cm-1 region in Figure 6.6 is striking, particularly considering the different structures and alkali metals involved. As indicated in Figure 6.3, peaks in this region are attributable in large measure to the tetrahedral alkali ion optic modes, and given the crystal structures, may therefore be anticipated in all the spectra shown in Figure 6.6, bearing in mind the impurities present (Table 6.1).

In bct Rb4C60 and the body-centred orthorhombic Rb4C60(ND3)2 phase present in the + "Rb3C60(ND3)2.5" sample, the Rb ions are in distorted tetrahedral sites, so that we may expect alkali optic modes at similar energies in these compounds. The overall density of states appears to be at somewhat higher energies in the "Rb3C60" sample than in the other rubidium fullerides, which is consistent with the higher force constants expected in the relatively congested tetrahedral sites of fcc Rb3C60. The quality of the data is such that we should not pursue this point too far, however. Raman studies of this energy region for the fcc fullerides at room temperature show a shift in frequency from ~80 to -1 ~120 cm on going from Rb3C60 through K3C60 to Na3C60, as expected from the relative alkali ion masses29.

6-9 Interpretation of the low energy peaks observed in "Rb3C60(ND3)2.5" is particularly difficult owing to the partial disproportionation (§ 4.1.5) into body-centred orthorhombic Rb4C60(ND3)2 and polymeric Rb1C60, with very poorly crystalline fcc

Rb3C60(ND3)4.8 remaining as the majority phase. The phase showing the sharpest diffraction peaks, Rb4C60(ND3)2 (25 wt%) has a structure which may be loosely described as an orientationally ordered version of Rb4C60 in which pairs of Rb atoms around the equator of the body-centred unit cell are bridged by two ND3 ligands. Its alkali optic vibrational DOS should therefore be comparable to that of Rb4C60. Because of its polymeric structure, the low energy DOS of the Rb1C60 phase can be expected to differ from that of the other fulleride phases, though in a way that has yet to be determined. The Rb3C60(ND3)4.8 lattice dynamics should be similar to those of Rb3C60, but because of the relatively large ND3 content and the dominance of this phase, we can expect a significant contribution to the DOS from motions involving the ND3 ligand.

Our previous studies of NH3 and ND3 dynamics in the Rb3C60(NH3)x/(ND3)x system, at a time when the structure of the ammoniated material was unknown, were analyzed in 30 the framework of Rb3C60 lattice vibrations and likely metal-ammonia motions . There, evidence was found for ammonia torsional vibrations at 60 and 131 cm-1 in the deuterated compounds, which in conjunction with other data was found to be consistent with a three-fold torsional potential with a 97 cm-1 barrier height. The poorly crystalline

Rb3C60(ND3)4.8 phase, which would account for the bulk of the inelastic scattering, has a structure not inconsistent with three-fold ammonia torsional potentials27 (the structure of

Rb4C60(ND3)2 dictates a six-fold potential, see § 4.1.5). Despite this, the -1 Rb3C60(ND3)2.5 data in Figure 6.6 shows peaks at 32, 145 and 161 cm , not 60 and 131 cm-1 as observed previously. Such changes are probably related to gradual sample aging31.

6-10 6.2 Intramolecular Vibrational Spectra

Measurements of the intramolecular vibrational spectra of the fullerides provide the most compelling evidence in support of the phonon-mediated superconductivity mechanism in the fullerides. We present here an overview of the recent literature regarding the C60 intramolecular modes in the fullerides and further neutron scattering results obtained in the present study using the TFXA instrument (see § 8.2.2).

6.2.1 Comparison with Optical Spectra

The C60 molecule has a total of 174 vibrational modes, divided into 46 classes according to their symmetries in the icosahedral point group (Ih) as follows:

Γvib = 2Ag + 8Hg + 3T1g + 4T2g + 6Gg + Au + 4T1u + 5T2u + 6Gu + 7Hu.

Of these, only the four T1u modes are visible in infrared (IR) spectra, while the two Ag modes and the eight Hg modes are allowed in Raman spectra. By a fortunate coincidence it is the Hg modes that are generally believed to be involved in the fulleride superconductivity mechanism.

As discussed in § 8.2.2, the TFXA instrument effectively probes all points in the Brillouin zone simultaneously. In Raman and infrared studies the momentum exchange associated with the interaction of a photon with the sample is negligible, so that only the zone centre is measured. As a result, the dispersion that occurs throughout the Brillouin zone can result in greater broadening of the neutron spectra compared to the Raman or infrared spectra.

A number of calculations have been performed for the mode frequencies of the 22 32 33 34 35 36 37 38 uncharged, isolated C60 molecule . The most accurate of these is probably that in reference [22], for which the rms error in the prediction of the experimentally observed Raman and infrared active peaks is 2.1%. Table 6.2 summarizes the available Brillouin zone-centre data obtained by Raman, infrared and theoretical techniques for C60.

6-11 The dispersion of the intramolecular modes throughout the Brillouin zone has also been calculated39 (see Figure 6.7). This indicates dispersion of as much as ~25 cm-1 in the -1 modes below 450 cm , a notable exception being the Hu(1) mode. (While being a useful guide to the anticipated dispersion in C60, Figure 6.7 does not accurately predict mode energies; for example it shows a gap in the DOS from 1400 - 1500 cm-1, contrary to experiment.) There is a growing number of lattice dynamics calculations of relevance to the intramolecular region for the doped fullerides also20 22 23.

Figure 6.7: Phonon-dispersion curves along some high-symmetry directions 39 for low temperature Pa 3 C60 (taken in modified form from Yu et al. ).

40 An excellent compilation of C60 Raman data is given by Horoyski et al. , while perhaps the best infrared data are those of Homes et al.41. There is a great abundance of Raman

6-12 and infrared fulleride data in the literature, the majority of which concerns the potassium compounds, though the intramolecular vibrational spectra of fullerides are largely independent of the alkali cation42. Table 6.2, Table 6.3, Table 6.4 and Table 6.5 summarize the available experimental and computational zone-centre data for C60,

K4C60, K3C60 and K6C60. In these tables the symmetries correspond to the calculated energies, while the energy columns show our assigned experimental values, which may occasionally differ from those made by the original authors. There does not appear to be

2- any data appropriate to C60 intramolecular vibrations in C60 salts at the time of writing.

We are interested in the changes that occur in the C60 molecule's vibrational spectrum upon formation of the molecular fulleride salts. A number of factors are involved: 1. Vibrational frequencies can be expected to change somewhat on reduction from

the neutral C60 molecule to the anions. 2. Crystal field splitting, including the factor-group splitting, and its variation throughout the Brillouin zone.

3. Symmetry lowering by Jahn-Teller distortions of the C60 anions would lead to the activation of modes that are otherwise silent in infrared or Raman spectroscopies. 4. Combination or overtone bands, which can also involve the intermolecular modes discussed in § 6.1. 5. Electron-phonon coupling, the phenomenon of primary interest in the context of phonon-mediated superconductivity. Collectively, these factors cause broadening, shifts and splitting of the observed peaks.

43 In undoped C60 above 258 K (the temperature above which molecules rotate freely ), most of these factors do not operate, since a rapidly spinning molecule will (on average) perceive itself to be in an isotropic field (though note that the period of even a low frequency vibration (202 cm-1 Þ τ = 1.7 x 10-13 s) is small compared to the 44 -11 reorientational correlation time at 300 K (0.9 x 10 s). Also, undoped C60 has no t1u- derived conduction electrons to which vibrational modes might otherwise couple, and for the same reason potential Jahn-Teller geometrical distortions are not an issue.

6-13 -1 Table 6.2: Zone-centre energies of C60 (energies in cm )

Symmetry Calc.22 Observed Symmetry Calc.22 Observed Hg(1) 270.2 262, Gu(3) 790 264, T2g(3) 800 268, T1g(2) 826 272, Gu(4) 937 47 274, T2u(3) 945 956 275, Au 973 45 276 Gg(4) 1037 25 3 T2u(1) 343 344 Hg(5) 1099 1102 25 Gu(1) 348 355 T2u(4) 1131 25 47 Hu(1) 388 404 Hu(5) 1176 1177 Hg(2) 431 429, T1u(3) 1183 1182, 4353 1183, 46 Gg(1) 480 485 1184, 3 41 Ag(1) 493 498 1185 T1u(1) 527 525, T1g(3) 1241 3 526, Hg(6) 1248 1253 47 527, Gu(5) 1259 1306 41 528 T2g(4) 1277 47 Hu(2) 527 527 Gg(5) 1287 47 T2g(1) 543 Hu(6) 1291 1331 47 T1g(1) 563 Gu(6) 1420 1425 Gg(2) 570 Hg(7) 1426 1426, 41 3 T1u(2) 576 576 1428 25 Hu(3) 661 673 T1u(4) 1429 1428, 3 Hg(3) 709 710 1429, 47 T2u(2) 725 732 1431, 47 41 Hu(4) 750 761 1432 47 3 Gu(2) 756 761 Ag(2) 1469 1470 Gg(3) 772 Gg(6) 1501 Hg(4) 773 772, T2u(5) 1546 3 47 775 Hu(7) 1566 1562 3 T2g(2) 788 Hg(8) 1573 1578

-1 Table 6.3: Experimental zone-centre energies of K4C60 (in cm ) (R denotes that the measurement was made on the rubidium compound)

Symmetry Observed Symmetry Observed 55 R 55 R Hg(2) 409 Hg(4) 742 48R T1u(1) 472 T1u(4) 1320, 51 52 T1u(2) 568 1351 55 R 52 Hg(3) 704 Ag(2) 1438

6-14 -1 Table 6.4: Zone-centre energies of K3C60 (energies in cm )

Symmetry Calc.22 Observed Symmetry Calc.22 Observed Hg(1) 252, 215, T2g(2) 772 258 244, Gu(3) 773, 262, 769 270, T2g(3) 793 49 275 T1g(2) 817 T2u(1) 344 Gu(4) 926, Gu(1) 360, 923 355 T2u(3) 930 Hu(1) 375, Au 936 375 Hg(5) 1019, 1089, 49 Hg(2) 404, 346, 1023 1121 407 386, Gg(4) 1049, 404, 1049 414, Hu(5) 1077, 42549 1074 Gg(1) 471, T1u(3) 1113 49 473 Hg(6) 1136, 1222 3 Ag(1) 481 496 1136 Hu(2) 492, T2u(4) 1139 494 T1g(3) 1224 T1u(1) 504 Gu(5) 1253, T2g(1) 541 1248 T1g(1) 543 T2g(4) 1255 51 T1u(2) 562 572 Gg(5) 1280, Gg(2) 571, 1277 571 Hu(6) 1278, 49 Hg(3) 658, 665 1281 52 663 T1u(4) 1436 1360 Hu(3) 668, Gu(6) 1333, 668 1340 T2u(2) 681 Hg(7) 1349, 1327, 49 Hu(4) 702, 1348 1390 3 694 Ag(2) 1474 1446 Gg(3) 723, Gg(6) 1490, 732 1487 Hg(4) 737, 694, Hu(7) 1499, 740 710, 1502 722, Hg(8) 1532, 1476, 746, 1529 150849 49 758 T2u(5) 1550 Gu(2) 750, 755

6-15 -1 Table 6.5: Zone-centre energies of K6C60 (in cm )

Symmetry Calc.23 Observed Symmetry Calc. 23 Observed Hg(1) 266, 268, (Tg) 808 3 274 280 (Au) 926 (Au) 323 (Au) 940 (Tu) 342 (Tu) 949 (Tu) 363 (Tu) 1018 (Tu) 377 Hg(5) 1109, 1093, 3 (Eu) 378 1107 1122 Hg(2) 419, 419, (Ag) 1136 3 412 427 (Tg) 1140 (Tg) 461 (Eu) 1161 50 T1u(1) 466 467 (Tu) 1172 50 (Ag) 474 T1u(3) 1215 1182 (Eu) 477 (Tu) 1240 (Tu) 488 Hg(6) 1268, 1232, 3 3 Ag(1) 507 501 1264 1237 (Tg) 526 (Tg) 1298 (Tg) 537 (Eu) 1328 (Tg) 560 (Tu) 1332 (Ag) 562 (Tg) 1340 50 T1u(2) 571 565 (Au) 1345 (Tu) 626 (Tu) 1349 (Eu) 627 (Ag) 1352 Hg(3) 660, 656, (Tg) 1360 3 50 660 676 T1u(4) 1395 1342 3 (Tu) 666 Hg(7) 1414, 1384 (Tg) 666 1423 (Tu) 668 (Au) 1451 (Eu) 678 (Tu) 1453 (Ag) 692 (Tu) 1461 (Tu) 706 (Eu) 1461 3 (Au) 711 Ag(2) 1469 1431 (Tg) 740 (Tu) 1476 (Au) 743 (Tg) 1486 (Tu) 744 (Ag) 1491 Hg(4) 769, 728, Hg(8) 1498, 1474, 769 7603 1507 14813 (Tg) 792

6-16 6.2.1.1 Peak Shifts

Shifting of peaks can be significant; for example the infrared-active T1u(4) mode in -1 50 Rb6C60 is shifted down in energy by 89 cm relative to undoped C60 . In general, the filling of antibonding orbitals upon anion formation reduces the force constants between atoms on the ball, resulting in a lowering of vibrational frequencies. The frequency 51 shifts for the infrared active T1u modes have been well documented , the frequency 50 52 decrease of the T1u(4) mode being particularly pronounced , while the linear shift of 3 the intense, Raman-active Ag(2) mode has led to its use in the determination of C60 molecular valence in newly prepared fullerides53 54. While the general trend is for modes to shift to lower energies upon doping, this is not universal; for example the 55 energy of the Ag(1) mode appears to increase slightly upon doping . Nonlinear shifts are observed for the Hg(1), Hg(2) and Hg(4) modes. The shift, which correlates with 55 broadening, is maximum in Rb3C60 but is also substantial for Rb4C60 , and is strongest for the Hg(2) mode. It has been argued that correlation amongst the valence electrons -1 can account for a reduction in the Ag(2) mode frequency by as much as 30 cm if the electron-phonon coupling for this mode is weak56. Unfortunately such shifting is not readily quantifiable in our TFXA data, for reasons described in § 6.2.3.

6.2.1.2 Factor-Group Splitting Molecular modes that are degenerate for symmetry reasons may have this degeneracy removed in the solid state, where the C60 molecular symmetry is reduced to the site symmetry in the crystal, giving rise to a factor-group splitting of the energy levels. The treatment we give here is not intended to be exhaustive, but rather indicative of the nature and extent of the splitting, particularly amongst the Gg, Gu, Hu and Hg modes of icosahedral C60. The strong Coulombic forces that operate in the ionic fullerides are expected to lead to greater peak splitting in these salts than in undoped C60.

If the crystal symmetry is known and if C60 orientational disorder does not complicate matters, the Ih symmetry of the C60 molecule is effectively lowered to the symmetry of the point group corresponding to its location in the crystal, and there will be a consequent lowering of mode degeneracy. In Pa 3 C60 and Na2C60 there are four symmetrically inequivalent C60s per unit cell, while in the high temperature

6-17 orientationally disordered C60 phase, the fcc A3C60, bct A4C60 and the bcc A6C60 phases, the C60 molecules are symmetrically equivalent. The relevant symmetry arguments for the Pa3 and Im3 fullerides may be found in reference [57], the results of which are reproduced in Table 6.6.

Table 6.6: Zone-centre mode splitting in Pa3 and Im3 fullerides57

Icosahedron Im 3 Pa3 2Ag 2Ag 2Ag + 2Tg 3T1g 3Tg 3Tg + 3(Ag + Eg + 2Tg) 4T2g 4Tg 4Tg + 4(Ag + Eg + 2Tg) 6Gg 6Ag + 6Tg 6(Ag + Tg) + 6(Ag + Eg + 3Tg) 8Hg 8Eg + 8Tg 8(Ag + Tg) + 8(Ag + Eg + 4Tg) 1Au 1Au Au + Tu 4T1u 4Tu 4Tu + 4(Au + Eu + 2Tu) 5T2u 5Tu 5Tu + 5(Au + Eu + 2Tu) 6Gu 6Au + 6Tu 6(Au + Tu) + 6(Au + Eu + 3Tu) 7Hu 7Eu + 7Tu 7(Au + Tu) + 7(Au + Eu + 4Tu)

40 The factor-group splitting is necessary in the explanation of Pa3 C60 and Na2C60 (in 49 3 both of which the C60 is in sites of Th symmetry), fcc K3C60 , and bcc K6C60 spectra. From an experimental point of view, the selection rules that apply to optical spectroscopy (Raman and IR) are extremely useful in assigning peaks - studies of the polarization ratios have also helped assign the symmetries of modes after splitting42. In Raman and infrared spectra, this symmetry lowering can lead to the activation of modes that were formerly forbidden. The lowering of symmetry leads to the Raman activation of all gerade- modes in Pa 3 C60, and well resolved Raman studies of single crystal C60 have taken advantage of this fact in terms of a more complete assignment of the Raman spectrum40 46 58 59.

At the time of writing the most detailed Raman study of the splitting of modes in single 40 crystal Pa 3 C60 is that of Horoyski et al. . Other workers have reported similar results45. In these spectra, the observed peak manifolds arising from factor-group splitting of a particular mode may span as much as 14 cm-1. Splitting of this magnitude can be seen in the well-resolved Hg(1) peak in the C60 INS data of Coulombeau in Figure 6.9 and is discussed further in § 6.2.4.

6-18 Disorder in the sample leads to a variety of local environments for the C60 molecule, which will result in a broadening of peaks.

6.2.1.3 Jahn-Teller Symmetry Lowering Jahn-Teller lowering of the molecular symmetry in the fulleride anions can also be expected to lead to the activation of modes that would otherwise be infrared and Raman inactive. The fact that this has not been observed in those techniques, as well as the arguments presented in § 5, suggest that geometry changes due to Jahn-Teller distortions are not significant.

6.2.1.4 Combination and Overtone Bands In some spectroscopic techniques, particularly those which observe peaks above the -1 frequency of the highest energy vibrational quantum (~1600 cm in C60), it is necessary to invoke the possibility of multiple quantum excitation. Such peaks have been 46 59 observed in high resolution single crystal C60 Raman measurements and complicate the analysis of the vibrational spectra. There is also evidence for them in the vibronic structure on the near-infrared electronic absorptions in § 5.3 (see Figure 5.6). Fortunately there is no need to invoke combination or overtone bands in the neutron results presented below.

6.2.1.5 Electron-Phonon Coupling

Numerous papers describe the broadening of C60's Raman active Hg modes upon formation of fullerides, attributing it to their coupling to the t1u conduction electrons. Asymmetric Breit-Wigner-Fano peak shapes60 have often been used to fit the broadened peaks, such a peak shape arising from the coupling of the vibrational transition to a continuous electronic transition excited by the incident laser radiation. Changes in the width of the Hg(2) mode above and below the superconducting Tc in K3C60 and Rb3C60 have been reported61, though the latter report appears not to have been substantiated by other workers.

At the time of writing the clearest data pertaining to the Hg(1) and Hg(2) modes are the 49 62 single crystal K3C60 results of Winter and Kuzmany . Their results (reproduced in

6-19 Figure 6.8) show that each of these two modes may be modelled using a set of five voigtians, whose intensity depends on the laser wavelength. This leads to the proposition that the degeneracy of the Hg modes has been lifted, each component

Figure 6.8: Observation of zone-centre dispersion of the Hg(1) and Hg(2) modes in single-crystal K3C60 at 80 K. The intensity distribution amongst the fitted peaks is a function of the laser wavelength. (From Winter and Kuzmany49)

coupling with a different strength to the t1u electrons. The approximately linear relationship between the widths of the fitted voigtians and their distance from the sharp high energy peak (the relevant energies are listed in Table 6.4) was found to be consistent with the interpretation that Hg(1) and Hg(2) intramolecular modes are involved in the dispersion of conduction electrons. Peak widths may be related to the electron-phonon coupling constant via the Allen equation (Equation 2.8, § 2.3.5) 63.

The Hg(1) and Hg(2) modes are also greatly broadened in Raman spectra of bct K4C60 55 and Rb4C60 , which are insulators. This observation adds weight to the general belief that the electron-phonon coupling mechanism is an on-ball effect in the fullerides, and is not dictated by the presence of a Fermi surface.

6-20 Splitting of the other Hg modes is also evident in the data of Winter and Kuzmany. The available data have been included in Table 6.4, though we believe it is possible that some of these peaks, particularly in the range 640 - 780 cm-1, are due to activation of formerly silent non-Hg modes by the factor-group splitting described earlier.

While the selection rules in Raman spectroscopy lead to a lack of spectral congestion, the same restrictions can lead to difficulties in making comparative studies of other modes. Also such optical spectra do not simply reflect the true vibrational DOS, the possibility of coupling to electronic transitions leading to further complications. Such difficulties may be overcome by INS studies.

6.2.2 TFXA Intramolecular Fulleride Data

Figure 6.9 shows the 200 - 2000 cm-1 energy region of the inelastic neutron scattering

(INS) spectra of C60, Na2C60, Rb3C60, Rb4C60 and Rb6C60, as recorded on the TFXA instrument (see § 8.2.2). This energy region covers the range of the vibrational modes of the C60 molecule. A general correlation between the peaks in the six samples is fairly clear, the correlation being worst for Rb6C60, where the neutron counting statistics are worst. The zone-centre energies tabulated in Table 6.2, Table 6.3, Table 6.4 and Table 6.5 are plotted beneath the respective traces in Figure 6.9.

Some precautionary remarks are necessary before describing the TFXA data in Figure 6.9. In the discussion that follows, we choose to divide the INS spectra into several regions where different levels of analysis are possible. Only in the well-resolved 200 - 450 cm-1 region will we attempt a quantitative analysis.

6-21 Rb6C60

Rb4C60

Rb3C60 Intensity

Na2C60

C60

2 3 4 5 6 7 8 9 2 1000 Energy Transfer (cm-1)

25 28 Figure 6.9: Inelastic neutron scattering spectra of C60 (data of Coulombeau et al. ), Na2C60, Rb3C60, Rb4C60 and Rb6C60 (data of Prassides et al. ). Zone-centre frequencies are marked with bars (red = Hg, green = T1u, blue = Ag, black = other) beneath the respective traces (see text).

6-22 6.2.3 Experimental Considerations

It is essential that the reader be aware of the limitations of the intramolecular TFXA data shown in Figure 6.9. The following aspects in particular have been addressed: 1. Counting statistics and instrumental resolution are restrictive. 2. Some shifting of the energy of the spectra occurs as a result of variations in the precise positioning and geometry of the samples. 3. Samples are generally not pure. 4. Kinematic (classical) scattering of light atoms by neutrons can potentially contribute to the scattering at high energy transfers.

Neutron counting statistics can be improved at the expense of the instrumental resolution by binning the data into coarser energy intervals. The data sets illustrated in Figure 6.9 were not rebinned in any way, and so best utilize the available resolution of the TFXA instrument. Since the raw data interval is an exponential function of the energy transfer, it is natural to display the data on a logarithmic energy scale. When plotted in this way, the instrumental resolution function (∆E/E ≈ 2%64) appears constant. 39 -1 The apparently dispersion-free Hu(1) mode (see Figure 6.7) at ~400 cm in the undoped C60 spectrum shows the instrumental resolution (the width of a gaussian fitted to this peak implies ∆E/E = 2.2%, see Table 6.7 in § 6.2.4). Inelastic neutron spectra of 15 65 C60 and Rb3C60 (see Figure 6.18) have been recorded on MARI (Multi-Angle Rotor Instrument) at ISIS, which has superior resolution to TFXA above 650 cm-1 and serves to complement the latter data sets in the 650 - 1600 cm-1 regions.

A quantitative estimate of the effect of the shift in a given spectrum due to uncertainty in sample positioning is not easily arrived at66, but awareness of the problem led to considerable care being taken in the alignment of our Na2C60, Rb4C60 and 67 Rb3C60(ND3)2.5 samples. Other workers have investigated the effect using non- fulleride samples; their results are reproduced in Figure 6.10, the lowest trace of which illustrates the extent to which the effect may be compensated by an empirical energy shift (the applied shift uses ∆E/E = constant, not ∆E = constant; recall the exponential data binning). It is clear that this kind of shift is not ideal, but the relatively small shifts

6-23 observed in the fulleride samples, and difficulties in binning the data in other ways (for example in quadratically increasing intervals) make it a reasonable approximation. Neutron Intensity

4 5 6 7 8 9 1000 Energy Transfer (cm-1) Figure 6.10: Inelastic neutron scattering spectra of hexamethylenetetramine with the sample in the correct position (top), and deliberately placed slightly forward of the correct position (middle). The bottom trace shows the residuals after subtracting a scaled and shifted (see text) middle trace from the top trace (least squares optimization).

Although less dramatic than the shifts seen in Figure 6.10, evidence for small instrumental shifts in the fulleride data may be seen by comparison of the Na2C60 and

C60 spectra, and also of the Rb3C60 and Rb4C60 spectra, particularly in the well-resolved 200 - 450 cm-1 region (see Figure 6.13 and Figure 6.14). Because of the considerable cross-contamination in these samples (Table 6.1), we expect some correlation between the relevant sets of spectra, which is best achieved by a slight downshift of the C60 data relative to Na2C60 (∆E/E = 0.01), and Rb3C60 relative to Rb4C60 (∆E/E = 0.015).

The Na2C60, Rb3C60 and Rb4C60 samples all contained very significant impurities, which were summarized earlier in § 6.1.3 (see Table 6.1). As described there, the sample cans also make a very small contribution to the scattering at the low energy end of the intramolecular region.

By shifting and subtracting the C60 spectrum from the "Na2C60" spectrum we can hope to eliminate the contribution of the C60 impurity in the latter; similarly the Rb3C60

6-24 impurity can be subtracted from the "Rb4C60" spectrum. Figure 6.11 shows the "pure"

Rb4C60 and Na2C60 spectra obtained by shifting the "Rb3C60" and C60 data sets to minimize the differences between them and the raw "Rb4C60" and "Na2C60" data, respectively, and performing the appropriate subtractions. Background functions were fitted to each trace prior to subtraction.

Rb4C60 Neutron Intensity

Na2C60

2 3 4 5 6 7 8 9 1000 Energy Transfer (cm-1)

Figure 6.11: Inelastic neutron scattering of "pure" Rb4C60 and Na2C60, obtained by subtraction of appropriate fractions of the "Rb3C60" and C60 data sets, respectively (see Table 6.1; C60 weight fraction = 0.25).

We are wary of a thorough analysis of these relatively "pure" spectra, however, for several reasons. Error bars (not shown) are greatly magnified by the subtraction procedure, and in the case of Na2C60 the appropriate amount of C60 to subtract is uncertain. The appearance of the "pure" spectra is found to depend fairly critically on the shift, and will be complicated by the different sample geometries employed (thicker sample cans can be expected to lead to broader peaks). The presence of C70 in the

"Rb3C60" sample may also lead to inaccuracy of the pure Rb4C60 spectrum in Figure

6.11. Because the essential features of the original "Rb4C60" and "Na2C60" spectra are preserved in Figure 6.11, we choose to base our discussion on raw spectra (Figure 6.9) rather than impurity subtracted "pure" spectra such as those in Figure 6.11.

6-25 The contribution from C70 or derived RbxC70 phases in the TFXA "Rb3C60" spectrum may be judged from Figure 6.12, where we plot the TFXA C70 data of Christidies et al.68 in the energy region of the quantitative fits in § 6.2.4. While there are no very distinct bands due to the C70 contamination, we believe that the broad weak -1 peak centred at 322 cm in the "Rb3C60" data is due to C70 contamination, which may -1 also mask details of the Hg(1) mode at ~260 cm and the Hu(1) and Hg(2) modes in the 370 - 450 cm-1 region. Neutron Intensity

200 250 300 350 400 450 Energy Transfer (cm-1)

Figure 6.12: Comparison of "Rb3C60" and undoped C70 data recorded using 68 the TFXA spectrometer. The C70 data is that of Christides et al. .

Recoil effects in the scattering atom lead to broad and more intense scattering at high energy transfers for hydrogenous samples. Amongst the spectra in Figure 6.9, the samples with the greatest hydrogen content are Rb3C60(ND3)2.5 and perhaps C60, the hydrogen being present as a minor impurity in the ND3 or C6D6 remaining from the preparation. The hydrogen content of the other samples is likely to be similar to that determined by other authors in fulleride samples examined by prompt gamma-ray 69 activation analysis , ie. approximately one H atom per two or three C60 molecules, and this is insignificant. It has not been necessary to invoke a kinematic scattering contribution for any sample.

6-26 6.2.4 TFXA data between 200 and 450 cm-1

Dispersion of the excitation frequencies throughout the Brillouin zone can be expected to affect inelastic scattering peak shapes when it is greater than the resolution of the spectrometer. To analyse this situation an instrumental gaussian resolution function has been fitted to each spectrum. Such fits confirm the need to use more than one gaussian -1 when fitting all but the ~400 cm Hu(1) peak in the C60 data set, but are generally very unstable, and lead to considerable difficulties in making assignments. For this reason, in

Figure 6.13 and Figure 6.14 we illustrate fits to the C60, Na2C60, Rb3C60, Rb4C60,

Rb3C60(ND3)2.5 and Rb6C60 data sets using a minimum number of gaussians whose width has been allowed to vary. For the purposes of fitting, the data in these figures have been rebinned in a constant energy interval, so that the usual linear energy scale is appropriate. The fitted parameters are summarized in Table 6.7; a linear baseline was employed in each case. Normalized peak areas for the region of the fits are also listed in order to assist peak assignments. In the following discussion the energies quoted are the fitted energies, and no attempt is made to account for the instrumental shifting described in § 6.2.3.

Although there is an obvious correlation between the strongest peaks in the spectra, assignment of the weaker peaks, which because of their breadth carry significant intensity, is not easy. The C60 data does not include any such weak features, and here the assignment is straightforward. This fact allows the areas under the C60 peaks to be used as references when attempting to assign peaks in the other spectra.

-1 The T2u(1) and Gu(1) modes in the 320 - 370 cm region are not theoretically expected to be strongly influenced by electron-phonon coupling. Two peaks are readily resolved -1 in the undoped C60 spectrum at 344 and 355 cm , with fitted widths ∆E/E = 2.8 and 2.7, respectively. This is larger than the instrumental resolution (∆E/E ≈ 2.2) and is believed to reflect the factor-group splitting. In the other spectra they cannot be readily resolved, and except in the case of Na2C60 the corresponding peak has been modelled using a single gaussian. In the Na2C60, Rb3C60 and Rb4C60 spectra the single peaks resulting -1 from the overlap of the T2u(1) and Gu(1) modes at ~350 cm have widths comparable to the overlapping pair of peaks in C60 (see Figure 6.9), indicating that the

6-27 C60 Rb6C60

Na2C60

Figure 6.13: Gaussian fits to TFXA data in the energy range

corresponding to the low energy intramolecular C60 modes; top 15 left: C60 , left: Na2C60, above: Rb6C60. The average size of the error bars associated with data points (2σ) is shown in the

top left of each figure. The Na2C60 sample contains significant

C60 impurity (see Table 6.1).

6-28 Rb3C60 Rb3C60(ND3)2.5

Rb4C60

Figure 6.14: Gaussian fits to TFXA data in the energy range

corresponding to the low energy intramolecular C60 modes; top

left: Rb3C60, left: Rb4C60, above: Rb3C60(NH3)~2.5. The average size of the error bars associated with data points (2σ) is shown in the top left of each figure. No sample is phase pure (see Table 6.1).

6-29 Table 6.7: Gaussian fits to TFXA data (200 - 450 cm-1)

C60 Na2C60 E ∆E/E Norm. Assign- E ∆E/E Norm. Assign- (cm-1) (%) Area ment (cm-1) (%) Area ment 267 4.4 0.27 Hg(1) 245 6.4 0.08 Hg(1) 344 2.8 0.14 T2u(1) 266 5.5 0.21 Hg(1) 355 2.7 0.18 Gu(1) 288 (5.8) 0.05 Al 404 2.2 0.23 Hu(1) 318 (11.1) 0.05 (bkg.?) 432 2.8 0.18 Hg(2) 344 4.4 0.20 T2u(1)/Gu(1) * 353 2.0 0.05 T2u(1)/Gu(1)

382 9.8 0.13 Hg(2) 392 2.2 0.06 Hu(1) * 400 2.0 0.08 Hu(1)

425 3.6 0.09 Hg(2)

Rb3C60 Rb4C60 E ∆E/E Norm. Assign- E ∆E/E Norm. Assign- (cm-1) (%) Area ment (cm-1) (%) Area ment 227 5.2 0.04 V 258 8.9 0.21 Hg(1) 258 9.7 0.09 Hg(1) 266 2.0 0.05 Hg(1) 270 3.6 0.10 Hg(1) 298 12.5 - Al/?? 322 8.9 0.08 C70 347 4.1 0.36 T2u(1)/Gu(1) 353 4.8 0.33 T2u(1)/Gu(1) 378 3.8 0.14 Hg(2) 378 2.3 0.06 Hg(2) 391 2.3 0.15 Hu(1) 398 5.0 0.23 Hu(1) 409 7.2 0.09 Hg(2) 427 4.4 0.07 Hg(2)

Rb3C60(ND3)2.5 Rb6C60 E ∆E/E Norm. Assign- E ∆E/E Norm. Assign- (cm-1) (%) Area ment (cm-1) (%) Area ment

244 13.6 - V/Hg(1)/ 273 3.6 0.14 Hg(1) δ N-Rb-N 352 16.4 0.65 T2u(1)/Gu(1) * 266 8.5 0.29 V/Hg(1)/ 375 2.0 0.10 Hu(1) δ N-Rb-N 427 2.5 0.11 Hg(2) 301 6.6 0.05 δ N-Rb-N 325 4.1 0.04 δ N-Rb-N 346 4.0 0.31 T2u(1)/Gu(1) * 376 2.0 0.03 Hg(2) 391 3.8 0.22 Hu(1) 416 4.0 0.06 Hg(2)

* These widths were constrained to the resolution of the TFXA instrument, ∆E/E ≈ 2%64.

6-30 factor-group splitting is probably not much larger in these fullerides than in C60. The corresponding peak is particularly broad in Rb6C60, but with better counting statistics would probably be resolvable into several quite widely separated components as predicted by the factor-group splitting (see Table 6.5 and the ticks under the Rb6C60 trace in Figure 6.9).

The other mode in the 200 - 450 cm-1 region not usually implicated in superconductivity -1 mechanisms is the Hu(1) mode at 404 cm in the C60 spectrum. As seen in Figure 6.9 it is possible that doping with alkali metals causes this peak to split into two components, -1 -1 at 378 and 398 cm in Rb3C60 and 378 and 391 cm in Rb4C60. On the other hand, calculations yield fairly small zone-centre splittings of the Hu(1) mode in K3C60 20 (Hu → Eu + Tu with zone-centre excitation energies of 398 + 405 , or 375 + 375 (ie. 22 -1 unsplit) ), so we tentatively assign the smaller peaks at 378 cm in the Rb3C60 and

Rb4C60 data to the rather dramatic changes that occur in the Hg(2) mode (see below). The relative areas of peaks in the 370 - 450 cm-1 region lend further support to this assignment. The Hu(1) mode is of some interest in that one model of electron-phonon coupling predicts it to have a very significant width in the fullerides (HWHM = 151 cm-1)74; the present data clearly show that such a large broadening does not occur.

The intensity distribution of C60's Hg(1) mode has been modelled as a single gaussian in the C60 panel of Figure 6.13 in order to better define the average energy, but it is in fact rather better fitted using two peaks, as noted by Coulombeau et al.15 25 - the resolution into two peaks can be anticipated by inspection of the unbinned data in Figure 6.9. The latter authors accounted for this as an accidental mode degeneracy contradicting available theoretical calculations, though here we propose that it is due to factor-group splitting and dispersion of the mode throughout the Brillouin zone (see Figure 6.7, the ticks beneath this peak in Figure 6.9, and Table 6.2).

-1 In the discussion of the fullerides' two Hg modes in the 200 - 450 cm region it is crucial to bear in mind the contributions from vanadium or aluminium sample cells (see

§ 6.1.3) and also the C60, RbxC70 and Rb3C60 contamination in the "Na2C60", "Rb3C60" and "Rb4C60" samples, respectively (see Table 6.1 of § 6.1.3, and § 6.2.3).

6-31 We focus first on the Hg(1) mode and describe first the Rb4C60 data set, this having the best counting statistics. The asymmetry of this peak is noteworthy, the sharp high energy edge at about 280 cm-1 with a tail to lower energies being at first glance reminiscent of the Raman work of other authors described in § 6.2.1.5. This structure is preserved after attempted subtraction of the Rb3C60 component (see Figure 6.11).

Although less compelling and perhaps enhanced by a small contribution from RbxC70

(see Figure 6.12), the Rb3C60 appears to show a similar asymmetry. We note that the low energy tail is much less broad than is observed in the Raman work of Winter and 49 Kuzmany , which is represented by the red ticks beneath the Rb3C60 data in Figure 6.9. This reflects the fact that the neutron scattering is not coupled to electronic transitions in the same way as the Raman data (see § 6.2.1.5), the neutron data instead giving a more accurate rendering of the true vibrational DOS. Given the narrowness of the Hg(1) mode in the INS results, however, it could perhaps be argued that the peak width and shape are due entirely to factor-group splitting and dispersion in the Brillouin zone. In -1 Na2C60 the distinct bump at 245 cm may also be assigned as a component of the Hg(1) -1 mode. The peak at 273 cm in Rb6C60 is well modelled by a single gaussian despite its clear splitting into two components at the zone centre as seen in Raman spectra (see

Table 6.5 and the red ticks beneath the Rb6C60 trace in Figure 6.9).

-1 We are unable to assign the broad low peak centred at 298 cm in Rb4C60, but believe it is due at least partly to the aluminium of the sample cell (see Figure 6.4). Its fitted area

(Table 6.7) suggests it would best be assigned as a component of the Hu(1) mode, though this appears 93 cm-1 away and is in other respects not much different from the -1 Hu(1) mode in the other fullerides. Weak peaks in the 280 - 330 cm region in other -1 spectra are also difficult to assign; the broad weak feature at 318 cm in Na2C60 may simply reflect a somewhat nonlinear background, while weak peaks at 301 and 325 cm-1 30 in the Rb3C60(ND3)2.5 spectrum could perhaps be due to δ N-Rb-N bending motions .

The Hg(2) mode is of particular interest, as it shows the most dramatic changes -1 throughout the fulleride series. Raman data of C60 show two peaks at 429 and 435 cm , as indicated in Table 6.2 and shown by the red ticks beneath the C60 trace in Figure 6.9. The fitted width of the single TFXA peak (2.8%) is wider than the instrumental resolution, supporting the Raman observation of factor-group splitting. A similar peak

6-32 is observed in Rb6C60, and also in "Na2C60", though note that the attempted subtraction of the C60 impurity from the latter spectrum (Figure 6.11) reduces its relative intensity. -1 A relatively weak peak is also observed between 400 and 450 cm in the Rb3C60,

Rb4C60 and Rb6C60 spectra.

As already described, in all save the C60 and Rb6C60 spectra there is a peak at slightly lower energy transfers than the Hu(1) mode, which we also assign to the Hg(2) mode. In each spectrum the two fitted Hg(2) peaks have widths quite comparable to other modes in the 200 - 450 cm-1 region, and perhaps contrary to expectation, are not unusually broad in the superconductor Rb3C60. (Observe also from Figure 6.12 that any contribution from C70 to the Rb3C60 peaks in the Hg(2) region is essentially "out of phase", which if anything would lead to some broadening as well as an intensity increase of the fitted Rb3C60 peaks.) There are no significant temperature dependent changes to the Rb3C60 spectrum in this region on either side of the superconducting Tc, contrary to an early Raman scattering report61, though this may be due to insufficient counting statistics.

The Hg(2) mode appears in the expected position in both the C60 and Rb6C60 spectra.

This, and the structural similarities of C60 to Rb3C60, and of Rb6C60 to Rb4C60 (see Figure 2.4), leads us to believe that on-ball electronic effects are responsible for changes in the Hg modes in the insulating Na2C60 and Rb4C60 and metallic Rb3C60 compounds.

6.2.4.1 Electron-Phonon Coupling Ideally we would like to be able to evaluate electron-phonon coupling constants by the application of Allen's formula, as has been done by others for Raman data (§ 2.3.5 and

§ 6.2.1.5). The fact that each of the individual peaks used to describe the Hg(1) and

Hg(2) modes has a width comparable to the factor-group splitting/Brillouin zone dispersion in the non-Hg modes, as well as the experimental difficulties outlined in § 6.1.3 and § 6.2.3, lead us to believe that the available data cannot yield accurate values for the electron-phonon coupling constants. Nevertheless in Table 6.8 we tabulate positions of the centre of mass as well as peak separations for the six fitted TFXA data sets, from which it is evident that the Hg(2) mode suffers much more dramatic changes upon partial intercalation than the Hg(1) mode. Comparison of our fitted and assigned

6-33 peaks in Figure 6.13 and Figure 6.14 with the Raman spectra of Winter and Kuzmany49 in Figure 6.8 (whose peak centres correspond to the red ticks beneath the Rb3C60 trace in Figure 6.9) suggests that coupling constants derived from neutron results would be smaller than those obtained from Raman data.

Table 6.8: Hg(1) and Hg(2) peaks

Hg(1) mode Hg(2) mode Centre of mass Peak sepn. Centre of mass Peak sepn. Sample Label (cm-1) (cm-1) (cm-1) (cm-1) * * C60 267 12 432 12 Na2C60 260 21 400 43 Rb3C60 264 12 404 49 Rb4C60 259 8 390 31 Rb3C60(ND3)2.5 258 22 401 40 * * Rb6C60 273 10 427 11 * These values are FWHM values of the peaks quoted in Table 6.7

n- Interestingly, a recent computational study of the ground states of the C60 (n = 1 - 5) anions suggests that the Jahn-Teller distortion in all these anions occurs primarily along the Hg(2) mode coordinates, with the Hg(1) mode also making a significant contribution70.

By themselves, these modes lead to atomic displacements in the C60 molecule as shown in Figure 6.15. In either case it can be seen that the distortion is axial (consistent with the spectral interpretations in § 5), and leads to a reduction of the symmetry of the Ih point group to D5d (depending on the choice of the icosahedral coordinate system); in combination with other modes, the symmetry may be lowered still further.

6-34 Figure 6.15: Stereo pairs of C60 showing atomic displacements 35 corresponding to the Hg(1) and Hg(2) modes. (From Harter and Weeks )

It would be interesting to obtain intramolecular INS spectra of molecular A1C60 (eg.

5- 54 quenched CsC60) and compounds containing the C60 anion (eg. RbBa2C60 ), as well as the recently characterized C60 intercalation compounds containing noble gases in the fcc octahedral interstices71. As described earlier in the context of Raman data of other workers (§ 6.2.1.5), it appears that the broadening is not related to the presence or absence of conduction electrons, which would explain the fact that broadening of the

Hg(2) and Hg(1) modes may be observed in insulating Na2C60 and Rb4C60 as well as superconducting Rb3C60.

6.2.5 TFXA data between 450 and 600 cm-1

Between 450 and 600 cm-1 the density of vibrational states contributing to the scattering intensity allows only a qualitative interpretation of the neutron data, as may be seen by the ticks beneath the traces in Figure 6.9. Significant doping-induced changes can be seen in this region despite the fact that it contains none of the Hg modes usually implicated in the superconductivity mechanism of Varma72 and Schlüter73. It appears

6-35 -1 that the peak at ~480 cm in C60 (due primarily to the Gg(1) mode) essentially maintains its intensity and position in all the samples shown in Figure 6.9, while the two broad C60 peaks at ~530 and 570 cm-1 (each due to several modes) are significantly shifted and split in the Na2C60, Rb3C60 and Rb4C60 data sets. The infrared-active T1u modes in this region (especially the lower, T1u(1) mode) show a systematic shift to lower energies in the higher fullerides (green ticks in Figure 6.9), and probably contribute significantly to the scattering intensity due to the Gg(1) mode, particularly in Rb4C60 and perhaps also

Rb6C60.

While it has been suggested that molecular modes with non-Hg symmetry may be involved in the electron-phonon coupling via non-adiabatic mechanisms, calculations show that the relevant shifting and broadening is relatively small, typically around 10% 74 of that expected for the Hg modes . Because it contains no Hg modes, we feel that the doping-induced changes in this region are a good indicator of the extent to which crystal field dispersion and shifts due to anion charging can influence the spectra.

6.2.6 TFXA data between 600 and 900 cm-1

In the C60 data set, peaks in this region are sufficiently distinct to be able to make several assignments; ticks beneath this trace indicate that the peak at ~670 cm-1 is -1 almost certainly due to the Hu(3) mode. The strong peak at ~720 cm is due at least in part to the Hg(3) mode, probably with some contribution from the T2u(2) mode also.

The other mode in this region which can be assigned relatively reliably is the T1g(2) mode at ~840 cm-1. Between 730 and 820 cm-1 is a region where the density of contributing states is too high to allow much hope of detailed assignment, but which contains the Hg(4) mode.

In the Na2C60, Rb3C60 and Rb4C60 spectra a general correlation with the C60 spectrum can be seen, though details are generally smeared, probably due to larger factor-group splitting of the many modes. Part of the general smearing could be due to broadening by electron-phonon coupling of the Hg(3) and Hg(4) modes, but it would be folly to attempt a quantitative description of it with the available resolution and counting statistics. In 49 single crystal Raman studies of Rb3C60 at 80 K the Hg(3) and Hg(4) modes appear to

6-36 be split into about six peaks covering the region 650 - 780 cm-1, though the split components do not show the same broadening as the Hg(1) and Hg(2) modes. Red markers beneath the "Rb3C60" spectrum in Figure 6.9 show the positions of the split components.

6.2.7 TFXA data above 900 cm-1

This region represents the high energy end of the intramolecular vibrational spectrum, where the difficulties with instrumental resolution and counting statistics are most pronounced. Inspection of Figure 6.9 suggests that there is significant correlation between the C60, Na2C60, Rb3C60, Rb4C60 and perhaps even the Rb6C60 data sets, but that 15 65 only a very qualitative description is possible. For C60 and Rb3C60 (see below, Figure 6.18) much superior data have been recorded in this energy region using the MARI inelastic neutron scattering instrument at ISIS.

-1 A peak at ~960 cm , particularly prominent in the C60 and Rb4C60 data sets, is probably due to the Gu(4) and T2u(3) modes (see Table 6.2); the Au mode may also be partly contributing, though its low degeneracy ensures that its contribution will be relatively small. The same feature can be seen as a rather noisy bump in the Na2C60 data, and is arguably also present as a broad feature in the Rb3C60 data set.

-1 Another peak is centred at about 1100 cm , which is clearest in the C60, Na2C60 and

Rb6C60 spectra. Table 6.2 and the ticks beneath the C60 trace suggest that it is due in large measure to the Hg(5) mode, perhaps with some contribution from the Gg(4) and

T2u(4) modes, an assignment supported by the low dispersion of C60 modes in this region (see Figure 6.7) and the considerable width of this peak in the MARI C60 data of Coulombeau et al.15.

Next, centred at ~1200 cm-1 is a broad and intense peak, which we believe draws its intensity from the Hu(5), T1u(3), T1g(3) and Hg(6) modes. In both the TFXA and the

MARI C60 data, there is considerable overlap of this feature with another broad and -1 intense peak centred at ~1300 cm , which we propose is due to the Gu(5), T2g(4), Gg(5)

6-37 and Hu(6) modes. The same general features may be seen in the Na2C60 and Rb4C60 data, and perhaps also the Rb3C60 spectrum.

At ~1400 cm-1 there appears to be a weak minimum in all the spectra, which is much clearer in the C60 and Rb3C60 MARI data. Weak and unresolved features may be seen in all the TFXA data sets between 1400 and 1700 cm-1, the latter value being an upper -1 bound on the energy of C60's intramolecular modes. The clear peak at ~1450 cm in the

C60 data may be reasonably assigned to the Hg(7) and T1u(4) modes. Three small peaks -1 in the C60 MARI data at 1470, 1510 and 1530 cm may be tentatively assigned to the -1 Ag(2), Gg(6) and T2u(5) modes, respectively, while the final, strong peak at ~1580 cm is due to the overlap of the Hu(7) and Hg(8) modes.

6.2.7.1 Temperature Dependence

For the superconducting Rb3C60 compound, it is of considerable interest to know if there are changes in the intramolecular modes at temperatures above and below the superconducting Tc (29 K). Although the Rb3C60 data set was not originally recorded as part of this work, the relevant data were recorded by members of our own group, and have been made available for the present re-analysis.

As described beneath Table 6.1, the Rb3C60 data presented in Figure 6.9 represents the summation of three data sets, one at 22.4 K, another at 35 K, and another covering the range 30 - 112 K (the superconducting Tc is ~29 K). In Figure 6.16 we re-present the data of reference [26] in the high energy region where it was proposed that temperature dependent changes may be occurring. We note that there is no obvious correlation between any of the spectra. Summation of two or more data sets and different binning can make particular features appear more or less significant, forming the basis of arguments presented in reference [26]. Because of the variation between the four data sets and in light of more recent MARI data, here we propose that the possible temperature dependent changes in the TFXA data are in fact due to statistical variations, and that none of the currently available data show convincing evidence of temperature dependence.

6-38 Intensity

1000 1200 1400 1600 Energy Transfer (cm-1) Figure 6.16: High energy TFXA inelastic neutron scattering data from Rb3C60 recorded at various temperatures. From bottom to top: 22.4 K, 35 K, 30 - 112 K and 147 - 290 K. Horizontal lines show the zero counts level for each data set. The superconducting Tc of Rb3C60 is 29 K.

Figure 6.17 and Figure 6.18 show Rb3C60 neutron scattering data for the high energy intramolecular modes at temperatures above and below the superconducting Tc (~29 K75) recorded using TFXA26 and MARI65, respectively. Data have been binned similarly in the two figures, which are shown on similar scales to facilitate comparison. The superiority of MARI for the examination of these high energy modes is obvious, despite difficulties encountered in the experiment65. The correlation between the TFXA and MARI data is in general quite poor, the common features being the peak at ~970 cm-1 and the general density of states in the region 1100 - 1600 cm-1, with a fairly sharp cutoff at the latter value.

6-39 2 Intensity (arb.)

1

1000 1200 1400 1600 Energy Transfer (cm-1)

Figure 6.17: High energy inelastic neutron scattering data from Rb3C60/70 as recorded on TFXA at ISIS26. Data showing error bars are the summation of two data sets (one recorded while cooling from 112 K to 30 K, the other at 35 K), while the solid line (error bars not shown, but of comparable magnitude) was recorded at 22.4 K. Rb3C60 has Tc ≈ 29 K.

Figure 6.18: High energy inelastic neutron scattering data from Rb3C60 measured on MARI at ISIS. The data showing the error bars were recorded at 20 K, the solid line representing data collected at 50 K. (Modified from Arai et al.65)

6-40 For the illustrated region, the Hg mode energies determined in Raman experiments of -1 K3C60 are at 1089, 1121, 1222, 1327, 1390, 1476 and 1508 cm (Table 6.4). While some of these values match the strong peaks in the MARI data, it must be noted that there are many other highly degenerate modes that are also contributing to the density of states in this region. At least in the MARI data, evidence for temperature dependent changes at the corresponding energies is not compelling.

In conclusion, we note that because the frequencies of the intramolecular modes is greater than the superconducting energy gap in the fullerides (ω > 2∆ ≈ 40 - 100 cm-1,76 77), we do not expect to see temperature dependent changes in neutron78 or Raman79 experiments, the intramolecular phonon energies being more than sufficient to cause pair-breaking at all temperatures.

6-41 6.3 References

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6-44 7. Conclusions

In this work we have probed the superconductivity response of the well-known face- centred cubic fulleride Rb3C60 to the intercalation of ammonia, both by gas-phase doping, and as a product from synthesis in liquid ammonia. Powder diffraction patterns from Rb3C60 made by vapour-phase intercalation and then exposed to gaseous ammonia are initially suggestive of icosahedral disorder. Several months' equilibration at room temperature allows the diffraction patterns to be fitted to a three phase mixture, containing in particular the previously unknown body-centred orthorhombic

Rb4C60(NH3)2, indicating that disproportionation has occurred. In the products from liquid ammonia preparations involving Rb:C60 ratios of 5:1, 4:1, 3:1 and 2:1,

Rb4C60(NH3)2 is consistently observed, and it may be the stability of this phase

5- 4- (supposedly an insulator like Rb4C60) that drives the disproportionation of the C60 , C60,

3- 2- 4- C60 and C60 anions present in solution to the C60 anion present in the solid state. The ammoniated RbxC60 materials are all non-superconductors.

Other new phases produced by synthesis in liquid ammonia have also been identified.

Body-centred cubic Li3C60(NH3)4 and Na3C60(NH3)6 have been identified and structurally characterized. We also discovered Na2C60(NH3)8, in which a previously unknown rhombohedral C60 stacking is observed, as well as a distinct phase of approximate composition Li~2C60(NH3)~8. Although Li~2C60(NH3)~8 powder diffraction data has proven difficult to model, approximate fits indicate that it also adopts a hitherto unknown C60 stacking pattern.

Deammoniation of these compounds by thermal desorption and subsequent annealing leads to structures that depend on the alkali metal involved. For Rb fulleride ammoniates the products were the same as those obtained directly from Rb and C60 in vapour-phase syntheses, the deammoniation approach leading to comparatively phase- pure samples. The C60 molecules in similarly deammoniated AxC60(NH3)y (A = Li or Na, x = 2 or 3, y = various) samples are predominantly hexagonal close-packed (hcp), but with a very high degree of stacking faults. Na2C60 prepared by direct vapour-phase

7-1 intercalation of Na into C60 shows some evidence of stacking faults, but much more closely reflects the face-centred cubic packing in the initial undoped C60. This dependence of the product on the preparative technique leads us to propose that the Pa 3 structure observed in Na2C60 by us and others is of comparable energy to an hcp structure, and it appears likely that the same conclusion applies to the Li fullerides also.

Considerable difficulties were encountered in measuring superconducting transition temperatures using the Low Field Microwave Absorption (LFMA) technique. To work around these, a technique to measure magnetic susceptibility transitions at radio frequencies (10 - 20 MHz) was developed. This technique was used to confirm that

Rb4C60(NH3)2 does not superconduct above 7 K, and also to establish that Na2C60(NH3)8 and Na3C60(NH3)6 are not superconductors above 7 K.

In connection with the synthetic work, a piece of apparatus for the spectrophotometric titration of solutions of alkali metals in liquid ammonia was developed. This enabled the observation of the first solution-phase near-infrared/visible/UV absorption spectra of

5- 6- 2- 3- the C60 and C60 anions, as well as considerable improvements to the known C60, C60

4- 1- and C60 spectra. Curiously, the well-known spectrum of the C60 anion could not be observed in liquid ammonia, though it is readily obtained in a wide variety of other

1- 2- solvents. By quenching 2-methyltetrahydrofuran solutions containing the C60, C60 and

3- C60 anions it was possible to generate transparent glasses which enabled modest resolution of the vibronic structure associated with the near-infrared electronic transitions. A plausible interpretation of the observed multiple near-infrared electronic origins involves Jahn-Teller distortions of comparable magnitude to the on-ball interelectronic repulsions in the ground and excited spin multiplet states that may be adopted by the various anions.

Inelastic neutron scattering (INS) measurements of Na2C60, Rb4C60 and disproportionated Rb3C60(ND3)2.5 were made, and analysed in conjunction with existing

Rb3C60 data recorded by other members of our group, as well as C60 and Rb6C60 data recorded by other workers. Despite spectral congestion arising from the absence of selection rules in INS, as well as difficulties arising from impure samples, the results are generally consistent with the computational, Raman and infrared studies of other

7-2 authors, in particular showing pronounced changes in the Hg(2) mode, and to a lesser extent the Hg(1) mode also. The observed changes are likely to be a consequence of electron-phonon coupling. Elsewhere the INS spectra showed considerable evidence for factor-group splitting, as well as some evidence for shifting of certain modes to lower energies as the anion charge is increased. We believe there is insufficient statistical evidence in the available INS data for the superconducting Rb3C60 compound to assert that temperature dependent changes occur in the scattering above and below the superconducting transition temperature, in either intermolecular or intramolecular energy regions.

7-3 8. Appendices

8.1 Superconducting Properties

Superconductivity was discovered by accident in 19111. It is characterized by immeasurably small electrical resistance and peculiar magnetic behaviour below a certain superconducting transition temperature, Tc. The purpose of this section is to outline the principal features of this behaviour. The electrical and magnetic properties both render the study of superconductivity a matter of considerable technological importance.

A property of superconductors that distinguishes them from a perfectly conducting metal is the Meissner effect; a superconductor is (almost) perfectly diamagnetic, expelling magnetic flux regardless of thermal or magnetic history, while a perfect (normal) electrical conductor would trap and retain the flux if suddenly brought into the perfectly conducting state while in a magnetic field2. The Meissner effect accounts for the sudden change in the magnetic susceptibility of superconducting samples observed in § 3.2.2 as the temperature is lowered through the superconducting Tc.

Nevertheless, if the magnetic field is sufficiently strong, the superconductivity is destroyed, with uniform penetration of the solid. The field strength required to cause this transition is dependent on the temperature, energetic considerations showing the critical field (Hc) to provide a direct measure of the relative energies of the superconducting and normal states. At the surface of the superconductor the magnetic field penetrates in an exponentially decaying fashion, the decay constant λ (the magnetic penetration depth) being characteristic of the superconducting compound.

The discontinuity of the heat capacity of a superconductor at the Tc indicates that the entropy of the superconducting state is lower than that of the normal (metallic) state. This, and the fact that the heat capacity transition is not accompanied by changes in the crystal structure, points towards an ordering of the conduction electrons below the Tc.

8-1 This ordering is associated with the fact that in a superconductor a mechanism exists which allows electrons close to the Fermi level to form pairs, known as Cooper pairs. Two such mechanisms have been proposed for the fullerides, discussed in § 2.3.4 and § 2.3.5. Single electrons, with S = ½, are fermions, and thus subject to the Pauli exclusion principle, which states that no two such particles can share the same set of quantum numbers. This accounts for the electronic filling of energy levels and bands in the molecular and solid cases as described in § 2.2 and § 5. On the other hand, the Cooper pairs are bosons, with S = 0, and the Pauli exclusion principle does not apply. A sufficiently high density of Cooper pairs allows the formation of a new coherent entity known as the condensate, which may be described by the wavefunction: Ψ= neiθ , (8.1) n being the density of pairs and θ a phase. The mathematical properties of the condensate account for the distinctive magnetic and electrical characteristics of superconductors.

The pairing of electrons near the Fermi energy leaves a gap in the electronic density of states, whose width, 2∆, can be measured by electron tunnelling3, infrared reflectivity4 and NMR spectroscopic5 techniques. For the fullerides, reported values lie in the range

2∆ = 3 - 5 kTc; a review of such results and the possible implications is given in reference [6].

The length describing the spatial extent of the Cooper pair, and which therefore dictates the density of pairs necessary for formation of the condensate, is the coherence length (ξ), and is characteristic of the particular superconductor.

Depending on the relative magnitude of the magnetic penetration depth λ and coherence length ξ, two kinds of superconducting behaviour are possible, known as type-I and type-II.

In type-I superconductors, ξ is greater than λ, which leads to a positive energy in the surface regions where the magnetic field penetrates the superconductor. The superconductor's energy may therefore be minimised by the complete exclusion of the magnetic field from the bulk of the material, or, for sufficiently strong fields and particular sample geometries, by the concentration of all the magnetic flux in a number

8-2 of small non-superconducting regions (a situation known as the intermediate state; see reference [7]). On the other hand, in type-II superconductors ξ is less than λ, which is associated with a negative surface energy for the penetration of magnetic fields, and for intermediate field strengths it is possible for quantized bundles of magnetic flux (called fluxoids) to penetrate the solid without destroying the superconductivity of the bulk material. The essential features of this behaviour are described by the magnetic field/temperature phase diagram shown in Figure 8.1.

Figure 8.1: Phase diagram describing the relationship between the normal, mixed, and Meissner states in type-II superconductors. (From Bishop8.)

If the fluxoid density is sufficiently high, the fluxoids will arrange themselves in a hexagonal two dimensional lattice. At lower densities the fluxoid lattice melts9 10 and may display glassy behaviour11 12. Although the investigation of such phenomena has not been undertaken as part of the present work, the fulleride superconductors are extreme type-II superconductors.

For type-II superconductors, values of λ and ξ may be estimated from measurements of 13 Hc1(0) and Hc2(0) using the following equations : æ h ö æλö H ()0 = ç ÷ lnç ÷, (8.2) c1è8πλe 2 ø è ξø h H ()0 = . (8.3) c2 4πξe 2

8-3 If a piece of superconducting material is broken in two, and the two pieces separated from each other by a sufficiently narrow gap (either vacuum or a non-superconducting material), then because of the tunnelling of Cooper pairs that may occur between them, the condensates of the two pieces will be neither identical nor completely independent. Such a gap is known as a "weak link", and has interesting properties that lead to a number of practical applications.

One phenomenon shown by weak links that was utilized in this work is the AC Josephson effect. The application of a potential difference V across such a junction will cause the tunnelling Cooper pairs to radiate photons of energy Dω = 2eV, and for voltages of the order of a microvolt the corresponding radiation falls in the microwave and infrared region of the electromagnetic spectrum. The converse effect also occurs, so that infrared or microwave radiation can excite currents in, and therefore be absorbed by, samples containing weak links. This fact is fundamental to one aspect of the Low Field Microwave Absorption (LFMA) technique described in § 3.2.1, in which the weak links are the numerous junctions between superconducting crystallites in the powder samples.

8.2 Overseas Facilities

Three major international scientific facilities were visited in the course of this work. A brief description of the facilities and the radiation they produce is followed by a more detailed description of the particular instruments that were used.

1. Argonne National Laboratory (ANL): The ANL in south-west Chicago is operated by the University of Chicago as part of the U.S. Department of Energy's national laboratory system. It is the site of the Intense Pulsed Neutron Source (IPNS) and a third-generation synchrotron, the Advanced Photon Source (APS) that is presently becoming available. The present work used the GPPD diffractometer (§ 8.2.1) at IPNS.

2. Rutherford Appleton Laboratory (RAL): This facility is a short distance from the city of Oxford in the UK. It houses the world’s brightest neutron spallation

8-4 source, ISIS, which is the source providing neutrons to the LOQ instrument14 and TFXA spectrometer (§ 8.2.2) used in this work.

3. Japanese National Laboratory for High Energy Physics (KEK): This centre, whose original purpose was to promote experimental studies on elementary particles, is located in Tsukuba, Japan. Facilities at this establishment include a 2.5 GeV linear accelerator that feeds the Photon Factory storage ring, a synchrotron source that provides X-rays for the Australian National Beamline Facility (§ 8.2.3).

At the time of writing, details of these facilities, their history, availability, capabilities and timetabling may be found on their World Wide Web sites15.

At ISIS, the production of particles energetic enough to produce efficient neutron spallation involves three stages. First, an ion source produces H- ions which are accelerated in a pre-injector column to 665 keV. In the second stage, a linear accelerator, the H- ions pass through four accelerating radiofrequency cavities to reach an energy of 70 MeV. Before injection into the third stage, the synchrotron, the electrons are stripped from H- ions by passage through a 0.25 µm alumina foil, producing a beam of protons. The proton synchrotron, 52 m in diameter, accelerates 2.5 × 1013 protons per pulse to 800 MeV, before they are extracted and directed towards a target of depleted uranium or tantalum. This process is then repeated at a rate of 50 Hz. The highly energetic protons produce neutrons by chipping nuclear fragments from the metal nucleus. For an 800 MeV proton beam impinging on a uranium target, around 25 neutrons are produced per proton. Although differing in technicalities, neutrons are essentially generated in the same way at IPNS.

Around the target, an array of small hydrogenous moderators exploit the large scattering cross-section of hydrogen to lower the energy of the very fast neutrons initially produced to the thermal energies appropriate for use in condensed matter studies. By repeated kinematic collisions with hydrogen nuclei the neutrons would eventually acquire a Maxwellian distribution if the moderator were made sufficiently large. The use of appropriately sized moderators at various temperatures thus allows the neutron energy

8-5 spectrum to be "tuned" for use in different types of instruments. Typical moderators are water (316 K, H2O), liquid methane (100 K, CH4) and liquid hydrogen (20 K, H2).

From an experimental point of view, the characteristics of neutrons produced in spallation sources such as at ISIS and IPNS have a number of features that distinguish them from neutrons produced in conventional reactor sources: 1. The pulsed nature of the source enables the use of time-of-flight techniques on white neutron beams, which allows fixed scattering geometries to be adopted. This generally allows greater versatility in terms of sample environment. 2. Neutrons from spallation sources have a relatively rich high-energy component in their energy spectrum, a feature useful for high resolution powder diffraction and high energy transfer inelastic scattering. Optimization of the flux at these energies is by the use of physically small moderators to preserve the sharpness of the initial neutron pulse (0.4 µs at ISIS). 3. The duty cycle of the accelerator ensures good signal-to-noise levels, as the source is essentially off when neutron scattering data are being collected.

Synchrotron radiation is produced by the sideways acceleration of charged particles (typically electrons or positrons) travelling at relativistic velocities. Relativistic velocities ensure good collimation of the radiation beam, as illustrated in Figure 8.2, so that essentially the entirety of the synchrotron radiation is directed towards a given target. The sideways acceleration is brought about by bending magnets (so called because they are also used to keep the particles orbiting within the synchrotron ring) - other "insertion devices" (undulators and wigglers) produce magnetic fields that alternate along the path of the charged particle but do not produce a net change of the direction of the particle beam, and can be designed to produce radiation with different characteristics. At the Australian National Beamline Facility (ANBF) a bending magnet is used to produce synchrotron radiation from the 360 mA beam of 2.5 GeV positrons typically used in the Photon Factory storage ring. The relative geometry of the synchrotron ring and the field of the bending magnet dictates that the synchrotron radiation is polarized with its electric vector in the plane of the synchrotron ring.

8-6 Figure 8.2: The sideways acceleration of electrons travelling at classical velocities results in a dipolar radiation field (case I), whilst at relativistic velocities the resulting radiation is well collimated (case II).

The synchrotron radiation energy spectrum is continuous, allowing a tunable energy X- ray source to be produced. Because of the way electrons/positrons are stored in the ring, the synchrotron radiation is pulsed, with a period that may be varied from 100 ns to 10 µs, with pulse widths of about 200 ps.

8.2.1 The General Purpose Powder Diffractometer (GPPD)

This instrument, housed at ANL, was used to collect neutron diffraction data for the disproportionated Rb3C60(ND3)2.5 sample of § 6, which is described by us in references [16] and [17]. The instrument itself is shown in Figure 8.3; neutrons from the moderator are directed down a 20 m flight path towards the sample, which was housed in an airtight cylindrical vanadium sample can. Most of the neutrons (λ = 0.5 - 5.0 Å) are scattered elastically, and are detected by four pairs of detector banks placed around the sample.

8-7 Figure 8.3: View of the General Purpose Powder Diffractometer (GPPD) at the Intense Pulsed Neutron Source (IPNS). The sample is positioned at the centre of the octagonal platform, which also houses the detectors. The flight tube coming from the spallation target may be seen in the background. (This image was downloaded from the IPNS web site15.)

8.2.2 The Time Focused Crystal Analyser (TFXA) Spectrometer

This instrument was used in the collection of data described in § 6. It differs from the other instruments used in this work in that it detects inelastically scattered radiation.

The general layout of the TFXA instrument is shown in Figure 8.4. Incoming neutrons are backscattered from the sample onto a pyrolytic graphite analyser, where they are Bragg scattered before being passed through a beryllium window and arriving at the detector. There are two detector tubes set up to detect neutrons scattered directly from the sample, enabling simultaneous collection of low resolution diffraction data.

8-8 Figure 8.4: The TFXA spectrometer. (Reproduced from reference [18].)

The purpose of the graphite analyser and beryllium window is to pass only those neutrons that leave the sample with a particular energy Ef. Bragg’s law: nλ = 2dsinθ (8.4) implies that the graphite will only reflect neutrons with multiples of a particular wavelength from the sample towards the detector tubes. In other words the only neutrons to reach the beryllium filter are those with energy hc nhc E= = . (8.5) f λ 2dsinθ The beryllium filter then allows only the lowest energy of these (corresponding to n = 1) to pass on to the detectors. In this way the energy of the neutrons reaching the detectors,

Ef, is determined by the fixed diffraction angle θ.

The quantity we need to know is the energy transferred to the sample by the inelastic scattering process. This is simply

Etrans = Ei(t) - Ef, (8.6) where Ei(t) is the energy of the incident neutrons, which (as we will show shortly) turns out to be a function of the detection time (t) only. Since Ef is fixed, the time of

8-9 detection of a neutron then corresponds to a particular energy transfer at the sample - hence the name of the instrument.

That Ei(t) is a function of time only is justified as follows. The time t taken for a neutron to get from the moderator to the detector (which is essentially the time since the initial neutron pulse at the source) is

t = ti + tf, (8.7) where ti is the time taken for the neutron to cover the distance li from the moderator to the sample, and tf is the time taken for it to cover the distance lf from the sample to the detector. If vi and vf are the neutron velocities before and after interaction with the sample, then the previous equation becomes: l l t= i + f . (8.8) v i v f The relationship between velocity and energy of the neutron (mass m), 2E v= , (8.9) m allows the expression for the time to be expanded further: m m + t=li l f . (8.10) 2E i 2E f Rearrangement then gives: ml 2 E = i (8.11) i ()2 2t - 2lff m 2E and since m, li, lf and Ef are all fixed and known, it is clear that Ei is a function of t only, as required.

The complexity of the expression just given for Ei and the uncertainties in t, li, lf and in particular Ef, will lead to a very complex expression for the energy (Etrans) resolution of TFXA, which is quoted in the instrument handbook as being simply ∆E/E ≈ 2%. As noted in § 6.2.3, this value agrees well with the width of the narrowest peak (Hu(1) 19 mode, fits to which imply ∆E/E = 2.12%) in the C60 TFXA data of Coulombeau et al. .

It is of interest to know which parts of the Brillouin zone are being sampled by the TFXA spectrometer for a particular sample. The relationship between energy and momentum transfer on the TFXA instrument is fixed, and given by:

8-10 E2Q≈ 2 , (8.12) where E is in meV and Q is in Å-1. Since each energy in TFXA is associated with a unique value of Q, and since the sample consists of a very large number of randomly arranged crystals, a given energy on a TFXA plot corresponds to measurement of the density of states on a spherical surface (radius Q) in reciprocal space. For fcc fullerides such as Na2C60 and Rb3C60 the lattice parameter is typically around 14.3 Å. The first allowed reflection (111) is the point closest to the origin in reciprocal space, and occurs at Q ~ 0.76 Å-1. The edge of the Brillouin zone is half this distance, ie. ~0.38 Å-1, which by Equation 8.12 corresponds to an energy of ~0.29 meV (2.3 cm-1). In other words, even for the lowest energies, TFXA's Q-sphere is well outside the first Brillouin zone, and the sampling of reciprocal space may be safely assumed to be isotropic.

8.2.3 The Australian National Beamline Facility (ANBF):

The experimental arrangement at Beamline 20a in Tsukuba is shown in Figure 8.5.

Figure 8.5: Layout of the Australian National Beamline Facility on Beamline 20a at Tsukuba, Japan. Radiation entering from the left is monochromated in the circular chamber in the centre, and passed to the Big Diff diffractometer (circular apparatus at right). (From Cookson et al.20.)

Monochromation of the incoming synchrotron radiation is achieved by diffraction from two successive water-cooled, parallel Si(111) faces. Rejection of higher harmonics is achieved by slight detuning of the second crystal (ie. making the crystal faces not quite parallel).

8-11 The Big Diff Diffractometer itself consists of a ~2.5 m diameter evacuable stainless steel cylinder that houses image plates 1 m away from the sample, which is mounted on the axis of the cylinder. The powder samples used in this work were loaded into 0.7 mm diameter Lindemann glass capillaries and sealed under argon with Torr-seal® cement. These were then mounted on a goniometer head, which was made to spin during data acquisition in order to more completely randomize the crystal orientation over the typically 5 minute exposure times. Image plates were then scanned and converted into intensity vs. angle data using the software available on location.

8-12 8.3 References

1 H. Kamerlingh Onnes, Akad. van Wetenschapen, Proc. Sect. Sci. (Amsterdam), 14, 113, 818 (1911) 2 "Introductory Solid State Physics", by H.P. Myers, ©1990 Taylor and Francis Ltd., London, UK. 3 P. Jess, U. Hubler, S. Behler, V. Thommen-Geiser, H.P. Lang and H.-J. Güntherodt, Synthetic Metals, 77, 201 (1996) 4 L.D. Rotter, Z. Schlesinger, J.P. McCauley Jr., N. Coustel, J.E. Fischer and A.B. Smith III, Nature, 355, 532 (1992) 5 R. Tycko, G. Dabbagh, M.J. Rosseinsky, D.W. Murphy, A.P. Ramirez and R.M. Fleming, Phys. Rev. Lett., 68, 1912 (1992) 6 C.M. Lieber, Z. Zhang, Sol. St. Phys., 48, 349 (1994) 7 "The Solid State", by H.M. Rosenberg, ©1988 Oxford University Press, Oxford, UK. 8 D.J. Bishop, Nature, 365, 394 (1993) 9 R. Cubitt, E.M. Forgan, G. Yang, S.L. Lee, D.McK. Paul, H.A. Mook, M. Yethiraj, P.H. Kes, T.W. Li, A.A. Menovsky, Z. Tarnawski and K. Mortensen, Nature, 365, 407 (1993) 10 M.F. Tai, G.F. Chang and M.W. Lee, Phys. Rev. B, 52, 1176 (1995) 11 D.M. Wang, R. Bramley and K.-P. Dinse, Physica C, 217, 16 (1993) 12 C.L. Lin, T. Mihalisin, N. Bykovetz, Q. Zhu and J.E. Fischer, Phys. Rev. B, 49, 4285 (1994) 13 K. Holczer and R.L. Whetten, Carbon, 30, 1261 (1992) 14 J.W. White, P.M. Saville, W.K. Fullagar and C.J. Hawker, ISIS 1995 Annual Report (RAL-TR-95- 050), A418 15 At the time of writing the internet addresses are: http://pnsjph.pns.anl.gov (Argonne National Laboratory), http://www.nd.rl.ac.uk (Rutherford Appleton Laboratory), and http://www.kek.jp (Photon Factory) 16 W.K. Fullagar, D. Cookson, J.W. Richardson Jr., P.A. Reynolds and J.W. White, Chem. Phys. Lett, 245, 102 (1995) 17 R. Durand, W.K. Fullagar, G. Lindsell, P.A. Reynolds and J.W. White, Mol. Phys., 86, 1 (1995) 18 User Guide to Experimental Facilities at ISIS, edited by B. Boland and S. Whapham (RAL-92- 041), December 1992 19 Coulombeau, H. Jobic, P. Bernier, C. Fabre, D. Schütz and A. Rassat, J. Phys. Chem., 96, 22 (1992) 20 D.J. Cookson, R.F Garrett, G.J. Foran, D.C. Creagh and S.W. Wilkins, Journal of the Japanese Society for Synchrotron Radiation Research, 6, 127 (1993)

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