test-typst/main.typ

@@ -82,18 +82,58 @@ In this paper, we do some things. We do something for the first time.
 
 = Results and Discussion
 
-// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
+== Phonons in Perfect 4H-SiC
-// 根据这些声子模式在拉曼实验中的表现，我们将这些声子分成三个部分。
+
+// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
+// 根据这些声子模式的极性，我们将这些声子分成两类。
+Raman scattering peeks correspond to phonons located near the $Gamma$ point in reciprocal space.
+We classified these phonons into two categories based on their polarities:
+  (i) negligible-polar phonons (i.e., phonons with no polarity or very weak polarity),
+    whose dispersion curves are continuous near the Γ point (as shown in @phonon),
+    and for which the effect of polarity can be ignored in the Raman scattering process;
+  and (ii) strong-polar phonons,
+    whose dispersion curves exhibit discontinuity near the Γ point (also shown in @phonon),
+    where the polarity gives rise to observable effects in the Raman spectra.
+
+#figure(
+  image("/画图/声子不连续/整体图.svg", width: 80%),
+  caption: [Weak- and None-polarized phonons near $Gamma$ point],
+  placement: none
+)<phonon>
+
+=== Phonons with Negligible Polarities
+
+// 我们使用 Gamma 点的声子模式来近似拉曼过程中的非极性声子。
+// 这个近似被广泛使用，并且由于这个原因而被认为是可行的：
+// 尽管拉曼过程中起作用的声子并不是那些严格在 Gamma 点的，
+//  但这些声子模式的散射谱在 Gamma 附近连续且导数为零，且波矢很小（在本文中大约 0.01 A，只有c轴的大约2%）。
+// 因此，它们的性质与 Gamma 点的声子模式区别不大。
+Phonons at the $Gamma$ point were used
+  to approximate negligible-polar phonons that participating in Raman processes.
+This approximation is widely adopted and justified by the fact that,
+  although the phonons participating in Raman processes are not these strictly located at the $Gamma$ point,
+  their dispersion near the $Gamma$ point is continuous with vanishing derivatives,
+  and their wavevector is very small
+    (about 0.01 AA in this paper, which corresponds to only 2% of the smallest reciprocal lattice vector of 4H-SiC).
+  // TODO: 确认这个数值
+Therefore, negligible-polar phonons involved in Raman processes
+  have nearly indistinguishable properties from those at the $Gamma$ point.
+
+// 4H-SiC 在 Gamma 有 21 个distinct phonons，其中极性较弱的有 18 个。这些模式对应于点群  $\mathrm{C_{6v}}$ 的 12 个表示（$\mathrm{3A_1+4B_1+3E_1+4E_2}$。
+// 对于非极性声子的拉曼散射，我们将使用 Gamma 点的声子模式来近似描述。
+// 这是因为，尽管拉曼实验中，起作用的声子并不严格在 Gamma 点；但这些声子模式的散射谱在 Gamma 附近是连续且导数为零，且起作用的声子离开 Gamma 点的距离也很小（对于 532 nm 的激光，离开 Gamma 点的距离小于 0.01 Å），它们的性质与 Gamma 点的声子模式区别不大。
+
+
+
+// 
+// 
+// 不同方向入射/散射光的拉曼实验对应于 Gamma 附近、偏向于不同方向的声子。
+// 其中 $\mathrm{A_1}$、$\mathrm{B_1}$ 为一维表示，对应于无简并的声子；$\mathrm{E_1}$、$\mathrm{E_2}$ 为二维表示，对应于二重简并的声子。
+// 在拉曼实验中，起作用的声子并不严格在 Gamma 点；但大多数声子的色散谱在 Gamma 点连续且导数（斜率）为零，因此大多情况下可以沿用这个分类，少数情况我们稍后会专门讨论。
+
+
+// 4H-SiC 在 Gamma 点的声子共有 21 个模式，这些模式对应于点群  $\mathrm{C_{6v}}$ 的 14 个表示（$\mathrm{3A_1+4B_1+3E_1+4E_2}$，其中 $\mathrm{A_1}$、$\mathrm{B_1}$ 为一维表示，对应于无简并的声子；$\mathrm{E_1}$、$\mathrm{E_2}$ 为二维表示，对应于二重简并的声子）。在拉曼实验中，起作用的声子并不严格在 Gamma 点；但大多数声子的色散谱在 Gamma 点连续且导数（斜率）为零，因此大多情况下可以沿用这个分类，少数情况我们稍后会专门讨论。
 
-Raman scattering peeks correspond to phonons located near $Gamma$ point in reciprocal space.
-We classified these phonons into three categories according to their behavior in Raman scattering:
-  (1) phonons could not be observed in Raman scattering spectrum,
-    either because they are Raman inactive or their scattering intensity is too weak;
-  (2) phonons could be observed in Raman scattering spectrum and with weak or no polarities,
-    their frequencies were independent of the direction of the incident light;
-  (3) strong polar phonons,
-    which were visible in Raman scattering spectrum,
-    and their frequencies depend on the direction of the incident light.
 
 // 我们计算了 4H-SiC 在 A-Gamma 和 Gamma-M 上的声子频率，如图和附录1所示。
 // 在拉曼散射中，起作用的模式都是那些非常接近于 Gamma 的模式