Nmr in Incommensurate and Chaotic Systems: the Detection of Solitons

Structural phase transitions may lead from a high temperature disordered phase with a space group Go and a unit cell edge a to: a) A ferrodistortive commensurate ordered state with a space group G1 and an unchanged unit cell, a' = a. b) An antiferrodistortive commensurate ordered state with a space group G2 and a unit cell size which is an integral multiple of the high temperature unit cell: a' = m.a, m = 2,3,4 ... c) An incommensurately modulated ordered phase where the wavelength X of the modulation is irrational in units of the original lattice constant a: — # M ; M, N = 1,2, a i\ 3,4 ... Here the translational symmetry is lost and the crystal cannot be described by any of the 230 three—dimensional crystallographic space groups though the system exhibits perfect long range order. d) A chaotic state with no long range order which can in contrast to amorphous systems — transform itself at lower temperatures to a commensurate ordered state. The transitions c) and d) take place when we have deterministic "competing interactions" and "frustration" (1); i.e. the competing structural units form a regular lattice. If the competing interactions are placed at random (such as in a mixed ferro—antiferroelectric crystal), a "spin glass" phase may be formed. One of the most interesting aspects of incommensurate phases is the spontaneous appearance of solitons (2) which are arranged in a periodic lattice with a period which is irrational in units of the original lattice. The soliton lattice may become chaotic thus leading to "chaos in space" as contrasted to "chaos in time" found in dynamical systems (3). The problems of the i) possible existence of only a few universal routes from order to chaos in dynamical systems, and ii) the non-linearity of the underlying natural phenomena have been extensively investigated in the last few years and are among the most outstanding open problems of condensed matter physics. The formation of solitons in spatial structures and the routes to spatial chaos have so far received considerably less attention. Both in dynamic and in spatial structures the soliton formation and the routes to chaos can be simulated by the onedimensional sineGordon equation (3) which can be written in dimensionless units as: