Commensurate-incommensurate phase transitions

Fig. 20. Phase diagram of the triangular lattice gas model with nearest-neighbor repulsion and next-nearest neighbor attraction, for JmJJ = — 1, in the coverage-temperature plane. For fl=0.5 a Kost tz-Thouless transition occurs at Ti and a commensurate-incommensurate transition at Tj. Two commensurate. 3 x 3 phases (with ideal coverages of 1/3 and 2/3, respectively) occur, as well as several two-phase regions, as indicated in the figure. Here l.g. stands for lattice gas and LI for lattice liquid . (From Landau. )...

In Fig. 13 the result of the field dependent calculation of the heat capacity using Landau theory for fields within the tetragonal plane is shown. In agreement with the experiment (Fig. 5), the magnetic field broadens the anomaly observed at the Neel temperature TN and lowers the transition point T of the commensurate to incommensurate phase transition. The increase of the calculated heat capacity while lowering the temperature below T, is a consequence of the approach to the temperature T of the proper instability of the ferromagnetic subsystem. As such, the same increase is observed in the experimental data too and it is possible to assume that the broad maximum at low temperature is connected with the subsequent phase transition in the magnetic subsystem of copper metaborate. 

Previously, sudden changes of amplitudes of magnetic satellites observable below 2 K [7] were interpreted in terms of lock-in transitions into phases with commensurate propagation vectors. Within the framework of the thermodynamic potential (5) it is impossible to describe a low-temperature transition from an incommensurate phase into a commensurate phase at zero external magnetic field. 

Figure 9. Model for understanding the commensurate-incommensurate phase transition. The Cu(A) system is ordered. The Cu(B) system is disordered because of the frustration caused by the geometrically competing interaction. The molecular field from the Cu(B) system disturbs the order in the Cu(A) system.

The Peierls169 metal-to-semiconductor phase transition in TTFP TCNQ p was detected in an oscillation camera these streaks became bona fide X-ray spots only below the phase transition temperature of 55 K this transition is incommensurate with the room-temperature crystal structure, due to its partial ionicity p 0.59, and the "freezing" of the concomitant itinerant charge density waves (this effect was missed by four-circle diffractometer experiments, which had been set to interrogate only the intense Bragg peaks of either the commensurate room-temperature metallic structure, or the commensurate low-temperature semiconducting structure). 

There is, however, another type of transition possible in two dimensions, a transition between states without LRO. This is the Kosterlitz-Thouless transition [8] mentioned in Sections II and V.B.l. It is relevant to superconductivity, commensurate-incommensurate transitions [61], planar magnetism, the electron gas system, and to many other systems in two dimensions. It involves vortices (thus the requirement of a two-component order parameter) characterized by a winding number q = (1/2-rr) dr V0, in which 0 is the phase of the order parameter (see also Ref. 4), the amplitude being fixed. These free vortices have an energy [see Eq. (28)] given by... 

Commensurability. Incommensurate lattice distortions and commensurate-incommensurate phase transitions are often observed in these materials. The incommensurability comes either from an incommensurate Fermi wave vector (2A F, 4kF scattering in charge-transfer salts) or from the counterion stacks (e.g., triiodide-containing materials). 

Higher order phase transitions in systems with two differently ordered commensurate phases or with a commensurate and an incommensurate phase have been suggested in 2D Meads UPD overlayers [3.93, 3.94, 3.110-3.114, 3.223]. However, there has not been clear experimental evidence supporting this assumption. 

Merwin LH, Sebald A, Seifert F (1989) The incommensurate-commensurate phase transition in akermanite, Ca2MgSi207, observed by in-situ Si MAS NMR spectroscopy. Phys Chem Min 16 752-756... 

Seifert F, Czank M, Simons B, Schmahl W (1987) A commensurate-incommensurate phase transition in iron-bearing Akermanites. Phys Chem Min 14 26-35... 

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

For any given irrational value of Q, there is an Q-dependent (dimensionless) threshold spring constant kc- Static friction vanishes for sufficiently strong springs k > kc- Below the critical value kc, metastability occurs and the static friction is finite. The transition from finite Fj to zero Fg is accompanied by a phase transition from a commensurate structure to an incommensurate structure [102,103]. For most values of Q and fixed, nonzero k and Vo, the winding number fi is a rational number near Q. Two neighboring intervals for which Q(Q) is a rational constant are separated by a point for which Q and are both irrational. The transition from commensurate to incommensurate is classified as a second-order transition [102,103], and consequently many properties are power laws as a function of k = k — kc, that is, for small K we have... 

A serious drawback of lattice gas models is their inadequacy to describe properly the commensurate - incommensurate phase transitions, often observed in real systems [144 - 150]. The possibility of the formation of incommensurate phases results directly from the finitness of potential berriers between adjacent potential minima and from the off-lattice motion of adsorbed particles. Although attempts have been made to extend the lattice-gas models and include the possibility of the formation of incommensurate solid phases [151,152], but it is commonly accepted (and intuitively obvious) that the continuous-space theories are much better suited to describe behaviour of adsorbed films exhibiting incommensurate phases. Theoretical calculations of the gas - solid potential for a variety of systems [88] have shown that, in most cases, the lateral corrugation is rather low. Nevertheless, it appears to have a very big influence on the behaviour of adsorbed layers. 

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