# 物理代写|光学代写Optics代考|EGR558

## 物理代写|光学代写Optics代考|Smectic-C – Smectic-A Phase Transition

The principal parameter that distinguishes smectic- $\mathrm{C}^$ from smectic-A is the tilt angle $\theta_0$. Because of the chiral character of the molecule, the tilt processes around the normal to the smectic layers, together with the transverse electric polarization $\vec{P}$ (cf. Figure 4.21). In the theories developed for describing phase transition from smectic-C to smectic-A phase, the tilt angle is treated as a primary order parameter of the system, very much as the director axis $\hat{n}$ in the nematic or cholesteric phase, while $\vec{P}$ is regarded as a secondary one.

Writing the two components of $\vec{P}$ and the tilt angle in the $x-y$ plane as $\vec{P}=\left(P_x, P_y\right)$ and $\left.\theta=\left(\theta_1, \theta_2\right)\right)$, the free-energy density $f_0(z)$ of the system can be expressed as a Landau type of expansion in terms of $\theta_1, \theta_2, P_x$, and $P_y$, in the following form $[48,49]$ :
\begin{aligned} f_0(z) &=\frac{1}{2} A\left(\theta_1^2+\theta_2^2\right)+\frac{1}{4} B\left(\theta_1^2+\theta_2^2\right)^2 \ &-\Lambda\left(\theta_1 \frac{d \theta_2}{d z}-\theta_2 \frac{d \theta_1}{d z}\right)+\frac{1}{2} K_3\left[\left(\frac{d \theta_1}{d z}\right)^2+\left(\frac{d \theta_2}{d z}\right)^2\right] \ &+\frac{1}{2 \varepsilon}\left(P_x^2+P_y^2\right)-\mu\left(P_x \frac{d \theta_1}{d z}+P_y \frac{d \theta_2}{d z}\right)+C\left(P_x \theta_2-P_y \theta_1\right) . \end{aligned}
In this expression, only the coefficient of the term quadratic in the primary parameter is temperature dependent, whereas the coefficient of the $P^2$ term is constant; this is so because it is not the interaction between the electric polarization that leads to a phase transition. The coefficient $A$ is of the form $A=A_0\left(T-T_{C A}\right)$, where $T_{C A}$ is the smectic-C-smectic-A transition temperature, $K_3$ is the elastic constant, and $\Lambda$ is the coefficient of the so-called Lifshitz term responsible for the helicoidal structure. $\mu$ and $C$ are the coefficients of the flexoelectric and piezoelectric bilinear couplings between the tilt and the polarization.

## 物理代写|光学代写Optics代考|ELECTROMAGNETIC FORMALISM

Scattering of light in a medium is caused by fluctuations of the optical dielectric constants $\delta \varepsilon(\vec{r}, t)$. In isotropic liquids, $\delta \varepsilon(\vec{r}, t)$ is mainly due to density fluctuations that are usually caused by temperature fluctuations but could also be due to some other mechanisms such as electrostriction and molecular reorientation (see Chapters 8 and 9). In the ordered phases of liquid crystals, an additional and important contribution to $\delta \varepsilon(\vec{r}, t)$ arises from fluctuations of the (birefringent) crystalline axis, lattice deformation, and flows.

These fluctuations or changes in the optical dielectric constant affect the spatial $(\boldsymbol{k})$ and temporal $(\omega)$ frequencies as well as the polarization state of the incident light. The direction, polarization, and spectrum of the scattered light are governed by the optical-liquid crystal interaction geometry.

In a uniaxial birefringent medium such as a liquid crystal, the dielectric constant tensor may be written as
$$\varepsilon_{\alpha \beta}=\varepsilon_{\perp} \delta_{\alpha \beta}+\left(\varepsilon_{|}-\varepsilon_{\perp}\right) n_\alpha n_\beta,$$
where $n_\alpha$ and $n_\beta$ are the components of a unit vector $\hat{n}$ (the director axis). Fluctuations in $\varepsilon_{\alpha \beta}$ come from changes in $\varepsilon_{\perp}$ and $\varepsilon_{/ /}$due to density and temperature fluctuations and from fluctuations in $\hat{n}[1]$.

## 物理代写|光学代写Optics代考|Smectic-C – Smectic-A Phase Transition

$$f_0(z)=\frac{1}{2} A\left(\theta_1^2+\theta_2^2\right)+\frac{1}{4} B\left(\theta_1^2+\theta_2^2\right)^2 \quad-\Lambda\left(\theta_1 \frac{d \theta_2}{d z}-\theta_2 \frac{d \theta_1}{d z}\right)+\frac{1}{2} K_3\left[\left(\frac{d \theta_1}{d z}\right)^2+\left(\frac{d \theta_2}{d z}\right)^2\right]$$

## 物理代写|光学代写Optics代考|ELECTROMAGNETIC FORMALISM

$$\varepsilon_{\alpha \beta}=\varepsilon_{\perp} \delta_{\alpha \beta}+\left(\varepsilon_{\mid}-\varepsilon_{\perp}\right) n_\alpha n_\beta,$$

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