Magnetic Fields Normal to the Helix Axis

In this section we will consider the magnetic field induced distortions for a field applied orthogonally (i.e., in the x, y plane) to the helix axis or z direction. For Ax 0, similar distortions to those discussed earlier will occur due to the director rotating in the field ft to minimize the free energy, i.e., with the helix in the x, y plane and n in the z direction. The distortions will be the same, but with slightly different boundary conditions depending on the pitch, cell thickness, and 

Initially, the sample is assumed to be fairly thick (i.e., d p) and in a planar texture. For zero or low fields, the helix will be arranged as in Fig. 33 a. At intermediate fields 

The field H is along 2 (i.e., H=h, 0, 0) and the director (see Eq. 1) is n = (cos y/, sin t/f, 0), where if=kz+. For thick cells in which / is a constant defined by the surface alignment and can be ignored. We will consider pitch deformations far away from the substrate. The free energy density of the system is then given by 

In this notation, a is a constant determined from the condition that 3 is a minimum, z=p H) 2, i.e., the half pitch of the field distorted structure or the distance between twist walls, (a) is a complete integral of the first kind, and 2 relates to the wall thickness , which is of the order of 2 2 (fF) [30, 139]. The free energy is then given by 

Here p2ia) is an elliptic integral of the second kind. Minimizing the free energy as in the previous sections for AS/3 j3=0 leads to 

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