Coherence length magnetic

This Section will present our first example of how a magnetic field can influence the orientation of the director n in a nematic liquid crystal. This example also introduces the idea of a magnetic coherence length. The equations presented here will be refashioned in a natural way, in the next Section, to obtain the equations required for the study of Preedericksz transitions. 

To this must be added the appropriate magnetic energy density. Since we are considering a semi-infinite sample, it is convenient in this situation to choose the form of magnetic energy provided by equation (2.108), giving 

The total energy per unit area in the xt/-plane is therefore 

The resulting Euler equation providing the equilibrium equation is given by (2.221), in this case reducing to 

Cleaxly, the boundary conditions (3.88) and (3.89) are all satisfied and the energy (3.83) becomes, after substituting for and nging (3.92), 

The geometry of this problem is shown in Fig. 4. A semi-infinite sample of nematic liquid crystal with positive anisotropy is bound on one 

Choosing the - sign for x 0 and integrating once more, we have 

[14] is plotted in Fig. 5. As we can see, is the characteristic length scale below which the magnetic field does not have much in- 

Suppose now that in place of the magnetic field an electric field is applied. If impurity conduction and other dynamical effects are neglected, the problem is qualitatively the same, and a quantity e can be obtained which is the exact analog of m Substitution of AeHw for Ax and E for H in Eq. [12] gives 

Therefore, one oersted of magnetic field is equivalent to the order of one volt/cm of electric field in terms of effectiveness in orienting the nematic liquid crystals. 

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The thickness of the Helfrich wail is of the order of 2, where is the magnetic coherence length defined as H k/Xs) (see 3.4.1), and the wall energy per unit length... 

Debye-Hfickel screening parameter observed, nuclear and magnetic widths of neutron diffraction lines superlattice period Bohr magneton effective de Geimes factor magnetic coherence length density of states... 

Therefore, the width of the individual peals is a measure of the number of bilayers that scatter coherently, or equivalently, the magnetic coherence length In... 

Fig. 25. Magnetic coherence length vs the inverse of the Y layer spacing. Dy layers are decoiqrled when the Ylayer thickness exceeds 140A.

In the presence of an external magnetic field it is convenient to introduce the magnetic coherence length... 

We define the magnetic coherence length as With tiiis definition (7.17) looks like ... 

Similar to the twist case, the magnetic coherence lengths can be defined for bend and splay deformations as ... 

This agrees with the sample thickness dependence as observed by Freedericks. From the definition of the magnetic coherence lengtii (see (7.20)) it is seen that, at the threshold for the Freedericks transition, the magnetic coherence length is i(Ff,i) =. ... 

Fig. 5—Tilt angle (deviation from the field direction) of the director field plotted as a function of distance from the wall. The distance is measured in units of magnetic coherence length, which is the characteristic iength of magnetic phenomena in nematic iiquid crystais.

The quantity H) is called the magnetic coherence length of the nematic. Taking as typical values (in Gaussian cgs units)... 

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