A Fiber Bundle Model of the Ice Ih Structure

A three-dimensional model of ice Ih based on the fiber bundle approach is presented. A hybrid structure of ice consisting of the oxygen lattice with the P63/mmc crystallographic symmetry and the hydrogen subsystem satisfying the Bernal-Fowler rules is considered. Controllable change of the protons position at hydrogen bounds by optoacoustic perturbation is discussed. Both classic and bispinor Hamiltonians are proposed. Forecasting intensive progress in microphysics as early as in 1959, R. Feynman has made a presentation at the annual session of the American Physical Society under the symbolic title ”There’s plenty of room at the bottom” [1]. He underlined that microcosm can give in future practically unlimited possibilities for material technology and information processing. But for achievement of practical results, it is necessary to overcome not a few obstacles. And one of them is the gap between microand macrolevels which prevents a direct contact without information losses and order distraction. Over the year, it was clearly understood that control and information theory should play an important role in microphysics progress as well as in molecular and quantum computing [2–7]. First of all, proper modelling of hybrid systems must be developed. One of possible models is presented here. Hexagonal ice Ih was chosen as an object thanks to its wide spread in nature and because it allows information processing at the molecular level. We start from the brief description of the usual ice structure following the short but very consistent book of N. Maeno [8]. Hexagonal modification of ice exists under normal pressure and temperature T < −6 ◦C. It has the crystallographic symmetry P63/mmc but only for the oxygen lattice, the hydrogen subsystem is characterized by relative arbitrariness in protons’ distribution at hydrogen bonds. So if the Bernal-Fowler rules are fulfilled, the protons have many variants of distribution. These rules are the next: 1) exactly two protons are situated beside every oxygen atom, 2) exactly one proton is present at every hydrogen bond. Mathematically, they can be expressed with some equations binding binary variables for bistable proton positions at hydrogen bonds. This and another technique will be demonstrated below. One of possible dispositions of protons in the frame of an ice Ih elementary cell is shown in Fig.1. Framework of the cell is composed of two mirror symmetric strata which are based on local orthogonal triplets e1, e2, e3 (lower,right) and e′1, e′2, e′3 (upper, left) with the origins situated at the endpoints of the vertical hydrogen bond. The triplets are oriented towards middle points of tetrahedron edges which are connected with the 280 Yu.I. Samoilenko Fig.1. correspondent oxygen atoms. Hydrogen bonds geometrically are represented by vectors fi, f ′ i (i = 0, 1, 2, 3), so that e1 = 1 2 (f1 + f0), e2 = 1 2 (f2 + f0), e3 = 1 2 (f3 + f0); e1 = − 1 2 (f ′ 1 + f ′ 0), e ′ 2 = − 1 2 (f ′ 2 + f ′ 0), e ′ 3 = − 1 2 (f ′ 3 + f ′ 0); f0 + f1 + f2 + f3 = 0, f ′ 0 + f ′ 1 + f ′ 2 + f ′ 3 = 0; e1 + e2 + e3 = f0 = f ′ 0 = −(e1 + e2 + e3); f1 = e1 − e2 − e3, f2 = −e1 + e2 − e3, f3 = −e1 − e2 + e3; f ′ 1 = −(e1 − e2 − e3) = − 5 3 e1 + 1 3 e2 + 1 3 e3 = − ( f1 + 2 3 f0 ) , f ′ 2 = −(−e1 + e2 − e3) = 1 3 e1 − 5 3 e2 + 1 3 e3 = − ( f2 + 2 3 f0 ) , f ′ 3 = −(−e1 − e2 + e3) = 1 3 e1 + 1 3 e2 − 5 3 e3 = − ( f3 + 2 3 f0 ) . Following the relations between vectors and having used the length of hydrogen bonds d = |fi| = ∣∣f ′ i ∣∣ = 0.276 nm, (i = 0, 1, 2, 3) one can determine horizontal a and vertical h moduli of the hexagonal lattice: a = |f2 − f3| = |a1| = |f3 − f1| = |a2| = 2 3 √ 6d = 0.452 nm, h = |a3| = |2f0 − f3 − f ′ 3| = 8 3 d = 0.736 nm. Proton configuration can be specified by different methods. One of them was applied in [6] to cubic ice Ic where protons’ positions have been determined by binary variables zα(n) ∈ GF (2) (α = 0, 1, 2, 3;n ∈ Z3). Displacement of protons from middles of hydrogen bonds could be expressed as follows: xα(n) = 1 2 (−1)α = ± 2 . A Fiber Bundle Model of the Ice Ih Structure 281 It was found that the Bernal-Fowler rules could be written in the form of homogenous equations