Fullerene Stability by Geometrical Thermodynamics

Abstract

This work proves that stability of C60 is a geometrical property of the thermodynamics of the system: a significant methodological advance since a detailed treatment of the energetics may be avoidable. This approach may be fruitful, not only for fullerenes but also for general problems of molecular stability and in other applications of conformational chemistry. For the non‐chiral C60, C384, and the weakly‐chiral C28, C76 and C380 (of these, C380 and C384 are classed as “unspirallable”), Schlegel projections are used to show that these fullerenes can all be represented by pairs of spirals counter‐propagating in anti‐parallel (C2) symmetry. For C60, the high symmetry is used to construct an analytical approximation for the spherical double‐spirals, shown mathematically to be Maximum Entropy (MaxEnt) using the formalism of Quantitative Geometrical Thermodynamics (QGT). Therefore C60 is necessarily stable. This MaxEnt stability criterion is general, depending only on the geometry and not the kinematics of the system. The sense and degree of chirality for C76 and C380 is also quantified using a Shannon entropy‐based fragmentation metric.

Halo Properties in Helium Nuclei From the Perspective of Geometrical Thermodynamics

“…Tables 1 & 2 confirm that the nuclear sizes calculated from QGT both of He isotopes and of the "helium series" are realistic. This, together with previous quantitative work by Parker & Jeynes [refs.1,16,17], indicates that this geometric entropy approach is valid for length scales over 35 orders of magnitude from sub-atomic to galactic.…”

“…The QGT approach can be extended to the nuclear matter radii of the "helium series" of nuclei, the self-conjugate A = 4n nuclei { 4 He, 8 Be, 12 C, 16 O, 20 Ne, 24 Mg, 28 Si, 32 S, 36 Ar, 40 Ca}. We find (for RMS matter radii calculated by QGT) that 8 Be should be to 4 He as 6 He is to 4 He.…”

Halo Properties in Helium Nuclei From the Perspective of Geometrical Thermodynamics

“…) Starting from the geometric entropy behaviour for a single particle, we can learn about larger systems with certain geometric symmetries (composed of multiple particles) from purely QGT and MaxEnt considerations. We have applied that logic to two simple sets of systems: the helium isotopes 4 He, 6 He and 8 He discussed above, and the self-conjugate A = 4n nuclei { 4 He, 8 Be, 12 C, 16 O, … 40 Ca} which we will call the "He series". Both series are constructed in the QGT model from individual nucleons, and their nuclear sizes are uniquely determined in both series from a single characteristic scale length given by the size of the proton and assumed to be the same for both systems.…”

“…PJ2019 also proved that the double-helix is a fundamental eigenvector of the entropic Hamiltonian, with its pitch and radius determining key properties. Separately, Parker & Jeynes 16 proved that a linear combination of two identical double-helices accounts for the stability of the spherical C60…”

“…(except the special case of 8 Be which is unbound). For convenience here, these nuclei are called the "helium series": 4 He, 8 Be, 12 C, 16 O, 20 Ne, 24 Mg, 28 Si, 32 S, 36 Ar, 40 Ca.…”