=== Phonons with Negligible Polarities

用 gamma 点的声子来近似。

Gamma 点的声子被用于近似参与拉曼散射的弱极性声子，无论入射配置。
这个近似是因为这些声子的色散曲线在 Gamma 点附近连续且非常接近 Gamma 点，并且已经被广泛使用。

Phonons at the #sym.Gamma point were employed to approximate negligible-polar phonons for all incidence configurations.
This approximation is based on the fact that
    the dispersion of these phonons is continuous and very close to the #sym.Gamma point,
  and has been widely adopted in the literature @_n-sic_2008.

18 个声子属于 12 个表示。拉曼张量的形状可以确定，大小不能。

这些声子的拉曼张量的形状可以通过对称性分析来确定。
18 个 Gamma 点的弱极性声子包含了 C#sub[6v] 点群的12 个不可约表示（2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]）。
通过进一步考虑 C#sub[6v] 中简并表示（E1 和 E2）在 C#sub[2v] 中的表示，所有声子的拉曼张量的非零分量可以确定，如表所示。
其中，B#sub[1] 模式具有零拉曼张量，不参与拉曼散射；
  其它表示的模式具有非零拉曼张量分量，可能可以在适当的偏振配置下在拉曼实验中观察到。
然而，模式是否足够强以在实验中可见取决于其拉曼张量分量的大小，而仅通过对称性分析无法确定这些大小。

The form of the Raman tensors (i.e., which components are non-zero) could be determined by symmetry analysis.
The 18 negligible-polar phonons at the #sym.Gamma point
  correspond to twelve irreducible representations of the C#sub[6v] point group
  (2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]).
By further decomposing the doubly degenerate modes (E#sub[1] and E#sub[2] of C#sub[6v] group)
    in the C#sub[2v] point group,
  the form of the 18 negligible-polar phonons' Raman tensors can be determined, as summarized in @table-rep.
Phonons of the B#sub[1] representation in C#sub[6v] possess zero Raman tensors
    and thus do not contribute to Raman scattering,
  while other phonons have non-zero Raman tensor components,
  making them potentially observable in Raman experiments under appropriate incidence and polarization configurations.
It should be noted, however, that the observability in Raman experiment depends not only on the tensor form,
  but also on the magnitude of its Raman tensor components,
  which cannot be inferred from symmetry considerations alone.

#include "table-rep.typ"

我们提出了一个新的办法来估计拉曼张量大小。

We propose a method to estimate the magnitudes of the Raman tensors of these phonons based on symmetry analysis.
This approach is founded on the assumption that the change in polarizability induced by atomic displacements in 4H-SiC
  is primarily determined by the first- and second-nearest neighbors of the atom and the sign of the atomic charge,
  while other factors (mass, bond length, etc.) only have small contributions.
Consequently, the Raman tensors of the calculated phonon modes can be estimated
    before additional first-principles computations (see appendix for details),
  and the results are summarized in @table-nopol.
The parameters $a_i$ exhibit significantly larger absolute values compared to $epsilon_i$, $eta_i$, and $zeta_i$,
  indicating the E#sub[2] mode at 756.25 cm#super[-1] in simulation (mode 8)
    possess a much higher Raman intensity than the others.

我们使用第一性原理计算得到了拉曼张量的大小，并与我们的结果进行了比较。

The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
  and the results are compared with both experimental data and theoretical predictions (@table-nopol).
The calculated phonon frequencies show good agreement with experimental data with a slight underestimation of 2-5%,
  where the error may be attributed to the underestimation of interatomic forces by the PBE functional (cite).
The calculated Raman tensors are also consistent with experimental and theoretical results.
Among negligible-polar modes, the E#sub[2] mode at 776 cm#super[-1] in experiment (mode 8)
    exhibits the highest Raman intensity,
  followed by four modes with lower intensities that are also experimentally visible,
  including the E#sub[2] modes at 195.5 cm#super[-1] (mode 1) and 203.3 cm#super[-1] (mode 2),
    the E#sub[1] mode at 269.7 cm#super[-1] (mode 3), and the A#sub[1] mode at 609.5 cm#super[-1] (mode 6).
The E#sub[1] mode calculated at 746.91 cm#super[-1] (mode 7)
    and the E#sub[2] mode calculated at 756.25 cm#super[-1] (mode 9)
  are predicted to have much weaker Raman intensities and are located close to the most intense mode (mode 8),
  making them indistinguishable in experimental spectra.
Additionally, the A#sub[1] mode calculated at 812.87 cm#super[-1] (mode 10)
  exhibits a very weak Raman intensity in the basal-plane polarized configurations (xx and yy, only 0.01)
  but shows an observable intensity when the polarization is along the z-axis (1.78).
Since most Raman experiments are performed in a back-scattering configuration with light incident along the z-direction
  (i.e., with in-plane polarization)
  and with photon energies much lower than the band gap,
  this mode is typically not observed (cite).
However, it should become detectable if the incident light has a polarization component along the z-direction
  (as in our experiment),
  or when the excitation wavelength approaches resonance conditions (cite).

其它峰在其它章节中解释。

Besides, there are other peeks in the experiment.
The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
Besides, there are small peeks at xxx,
  which could not be explained in perfect 4H-SiC and will be discussed in the next section.

// TODO: 将一部分 phonons 改为 phonon modes
// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。
// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode

#include "table-nopol.typ"

#include "figure-raman.typ"

// TODO: 解释为什么 E1 可以看到
