将经典的表示理论移动到附录

paper/appendix/default.typ

@@ -1,3 +1,4 @@
 = Appendix
 
+#include "representation/default.typ"
 #include "predict/default.typ"

paper/appendix/representation/default.typ

@@ -0,0 +1,18 @@
+== Symmetry Analysis of Phonons in 4H-SiC
+
+The phonons at the #sym.Gamma point in 4H-SiC (space group P6#sub[3]mc)
+  corresponds to four irreducible representations of C#sub[6v] point group
+  (4A#sub[1] + 4B#sub[1] + 4E#sub[1] + 4E#sub[2]),
+  including translational movement (1A#sub[1] + 1E#sub[1]), strong polar modes (1A#sub[1] + 1E#sub[1]),
+  and negligible-polar modes (2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]).
+The form of the Raman tensors for each representation is well known and summarized in @table-rep.
+
+#include "table-rep.typ"
+
+By further decomposing the doubly degenerate modes (E#sub[1] and E#sub[2] of C#sub[6v] group)
+  into the C#sub[2v] point group,
+  the raman scattering corresponding to vibration along x and y directions could be distinguished.
+For example, the E#sub[2] peeks observed in the Raman experiment of z(yy)#overline[z]
+    correspond to phonons vibrating along the y direction,
+  while the E#sub[2] peeks observed in z(yx)#overline[z] correspond to phonons vibrating along the x direction.
+This is important when the x and y directions become inequivalent and the double degeneracies are slightly lifted.

paper/result/perfect/non-polar/default.typ

@@ -1,42 +1,24 @@
 === Phonons with Negligible Polarities
 
-对弱极性声子的分析首先从理论出发，然后结合
+对弱极性声子的分析的传统理论分析。（Gamma 近似，C6v表示）
 
-对弱极性声子的理论分析，首先使用 Gamma 点的声子来近似，然后再讨论不同入射方向导致的差异。
-使用Gamma 点的声子来近似，基于这样的事实：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 @_n-sic_2008。
+对弱极性声子的理论分析，使用 Gamma 点的声子来近似。
+这基于这样的事实：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 @_n-sic_2008。
+对称性分析从理论上指出了这些声子属于 C6v 的 12 个表示。
+其中，B1 无活性，其它则有可能在拉曼实验中观察到。
+然而，模式的拉曼散射强弱无法仅从对称性分析中推断出来。
 
-Negligible-polar phonons were theoretically analyzed,
-  starting with the approximation using phonons at the #sym.Gamma point,
-  followed by a discussion of modifications arising from non-zero wavevectors.
+The negligible-polar phonons were theoretically analyzed by approximating their properties at the #sym.Gamma point.
 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。
-
-使用对称性分析来从理论上研究声子。
-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。
-
-Symmetry analysis was utilized
-  to theoretically investigate the properties of 18 negligible-polar phonons at the #sym.Gamma point.
-These phonons 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 form of Raman tensor,
-  but also on the magnitude of its Raman tensor components,
-  which cannot be inferred from symmetry considerations alone.
-
-#include "table-rep.typ"
+Symmetry analysis pointed out that
+  these phonons correspond to twelve irreducible representations of the C#sub[6v] point group
+  (2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2], see appendix for details).
+In which the B#sub[1] representation is Raman inactive,
+  while other phonons are potentially observable in Raman experiments.
+However, it should be noted that the intensity of Raman scattering from each mode
+  cannot be inferred from symmetry considerations alone.
 
 我们提出了一个新的办法来估计拉曼张量大小。
 
@@ -48,14 +30,14 @@ It should be noted, however, that the observability in Raman experiment depends
 我们的结果表明，E2-3 模式的拉曼散射强度远高于其它振动模式，这与实验和第一性原理计算结果一致。
 我们的研究表明，这个峰的高拉曼强度来自于所有键的贡献的相长干涉，这与其他弱极性模式不同（他们的贡献相互抵消）。
 
-A method to estimate the magnitudes of the Raman tensors of each mode from their vibration patterns (eigenvectors)
-  was proposed (see appendix for details).
+To overcome this limitation,
+  a method to estimate the magnitudes of the Raman tensors of each mode was proposed (see appendix for details).
 This approach was founded on the symmetry analysis and incorporates the assumption
   that the primary contribution from each atom to the Raman tensor arises from its nearest neighbors (denoted as $a_i$),
   while contributions from more distant atoms are much smaller (denoted as $epsilon_i$, $eta_i$, and $zeta_i$).
 Furthermore, the absolute amplitude differences among atoms of the same type within a phonon mode was neglected,
   and only their vibrational directions were considered.
-This enables a preliminary estimation of the Raman tensor magnitudes prior to detailed first-principles calculations,
+This enables a preliminary estimation of the Raman tensor magnitudes prior to expensive first-principles calculations,
   with the results summarized in @table-nopol.
 Our analysis gave the result that the E#sub[2]-3 mode
   should possess a much higher Raman scattering intensity than the others,