The Lennard-Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices

Abstract

Analytical formulas are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Grüneisen parameter within the Einstein model for the cubic lattices (sc, bcc, and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard-Jones and Ingham in 1924, Corner in 1939, and Wallace in 1965. The formulas are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulas give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.

Instability of the body-centered cubic lattice within the sticky hard sphere and Lennard-Jones model obtained from exact lattice summations

“…More importantly, the E * ELJ (A) curve does not change substantially in shape and is only slightly shifted compared to the (12,6) LJ potential, as shown in Figure 3. This is perhaps expected from the comparison between the two potentials (see appendix), and from the fact that for the fcc structure E * (R * min , 1.0) = 7.8532 [11] for the ELJ potential and close to E * (R * min , 12, 6, 1.0) = −L 2 6 /(2L 12 ) = 8.6102 for the (12,6) LJ potential (exp. E * = 6.4951 using the data from Ref.…”

“…The Einstein frequency ω E of a single atom moving in the field of all other atoms can be expressed analytically in terms of lattice sums [11]. ω E (a, b, A) > 0 for all A ∈ [ 1 3 , 1] and a > b > 3.…”

“…Solid-to-solid phase transitions are commonly modeled by computer intensive molecular dynamic or Monte-Carlo simulations at finite temperatures and pressures [2,3], or by various algorithms to find phase transition paths on a Born-Oppenheimer hypersurface [4]. For example, the relative stability of the fcc versus the hexagonal close packing (hcp) and possible transition mechanisms between these two phases for the rare gas elements has been a matter of a longstanding controversy [5][6][7][8][9][10][11]. While fcc has a higher excess entropy compared to hcp by a rather small difference (for the hard sphere model it is 0.00115±0.00004 k B per sphere [5]), the energetic stability of the fcc over the hcp phase for the rare gas solid argon (at low temperatures and pressures) is due to quantum effects (phonon dispersion) [8,10].…”

“…These lattice sums, which can be evaluated to computer precision, will be introduced in the next section and applied to analyse the energy profile of the bcc lattice distortion into the fcc densest packing using LJ and SHS interaction potentials. For more realistic two-body forces we apply extended Lennard-Jones potentials [26] for Ar 2 [11] and Cr 2 .…”

Stability and equation of state of face-centered cubic and hexagonal close packed phases of argon under pressure

“…Recent measurements of argon equation of state, using modern synchrotron techniques, focused on the very high pressure domain ( 40 GPa) 5 , 6 . The prediction of the structure adopted by rare gas solids is more difficult to address, because face-centered cubic and hexagonal close packed rare gas solids are almost isoenergetic, so that subtle effects such as many-body interactions, or phonon dispersion curves play a role in their relative stability 7 – 9 . Rare gases crystallize from the liquid under fcc structure, and it has been observed that compression favors the hcp phase in the case of xenon/krypton, with a sluggish transition observed or expected around 80 GPa/400 GPa at 300 K 10 – 12 .…”