Phenomenological theory of incommensurate phases in biphenyl

The crystal of biphenyl can exist in the monoclinic normal phase I, incommensurate stripelike phase II, and incommensurate phase III propagating along the twofold symmetry axis. The incommensurate phases II and III arise as a result of the condensation of modes of symmetry ${\Gamma }^{\mathrm{\{}\mathrm{k}\mathrm{\}}}$ and $T^{(\mathrm{k}}$,1) for the wave vectors from a general point of the reciprocal space and a point on the twofold symmetry axis, respectively. We write the phenomenological free-energy expansion in terms of the basic functions of ${\Gamma }^{\mathrm{\{}\mathrm{k}\mathrm{\}}}$ and describe both phases II and III. Hence, it follows that phase II proves to be a stripelike phase and not a quiltlike one. The gain of symmetry in the phase transition from phase II to phase III is introduced into the free energy by imposing those relations between the basic functions of ${\Gamma }^{\mathrm{\{}\mathrm{k}\mathrm{\}}}$ which follow from the projection of ${\Gamma }^{\mathrm{\{}\mathrm{k}\mathrm{\}}}$ onto $T^{(\mathrm{k}}$,1). We show that in the local anharmonic approximation the free energies of phases II and III, written in terms of the first modulation harmonic, are equal up to infinite order of invariants. Moreover, the same free energies remain equal if one includes contributions following from the second, third, and fourth modulation harmonics. We argue that the phase transition between these phases is not of a usual lock-in type. One can, however, consider phase III as a lock-in phase with the modulation wave vector k=(6/13. To find reasonable lock-in terms for this phase, higher-order anharmonic invariants should be introduced into the free-energy expansion.