test-typst/main.typ

@@ -34,6 +34,7 @@
   //   body
   // })
 )
+#set figure(placement: none)
 
 = Introduction
 
@@ -109,8 +110,7 @@ We classified these phonons into two categories based on their polarities:
     (b) Magnified view of the boxed region in (a).
       The orange dashed lines mark the phonon wavevectors involved in Raman scattering
         with incident light along the z- and y-directions.
-  ],
+  ]
-  placement: none
 )<phonon>
 
 === Phonons with Negligible Polarities
@@ -155,18 +155,39 @@ In these 18 modes, the vibrations of two Si atoms are approximately opposite to
   caption: [Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC.]
 )<bec>
 
+// 这18个声子对应于 $\mathrm{C_{6v}}$ 点群的 14 个表示：2A1 + 4B1 + 2E_1 + 4E2
+// 其中，B1 表示没有拉曼活性，它的拉曼张量为零；其它表示的拉曼张量不为零，但张量的大小是否足够大到可以在实验上看到，则还需要第一性原理计算，不能直接通过表示来判断。
+// 我们的计算结果如表所示。其中有几个声子的拉曼活性较弱，有几个比较强。强的都可以在实验上看到；但弱的能否看到则取决于它是否恰好位于强模式的附近。
+// 其中，xxx 和xxx 位于强模式的附近，它们在实验上无法看到；xxx 只在 z 方向入射/散射时可以看到；xxx 则在任意方向都能看到。
 
-// These phonon modes correspond to twelve irreducible representations of the $\mathrm{C_{6v}}$ point group: $\mathrm{3A_1 + 4B_1 + 3E_1 + 4E_2}$.
+The 18 negligible-polar phonons correspond to 14 irreducible representations of the C#sub[6v] point group:
+  2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
+Phonons belonging to A#sub[1] and B#sub[1] representations are non-degenerate,
+  while phonons belonging to E#sub[1] and E#sub[2] representations are doubly degenerate.
+Phonons belonging to B#sub[1] representation are Raman inactive, as their Raman tensors vanish.
+In contrast, the Raman tensors of phonons belonging to other representations have non-zero components,
+  indicating that these phonons might be visible in Raman experiment under appropriate polarization configurations.
+However, the actual visibility of each phonon depends on the magnitudes of its Raman tensor components,
+  which cannot be inferred solely from symmetry analysis.
 
+/*
+这里应该有办法来估计。下面是我总结的规律：
+按照我们规定的 ABCB 层序，并将拉曼张量的大小归结为键长的变化的话：
+* 对于 E2 表示（AC层运动方向必须相反，B1/B2层运动方向必须相反，因此只讨论A和B1层）
+  * A 层内部的那个竖的键，同向运动会导致比较大的拉曼张量
+  * B1 层内部的那个竖的键，反向运动会导致比较大的拉曼张量
+  * A 层和 B1 层之间的那个横的键，反向运动会导致比较大的拉曼张量
+我们或许可以通过这个路径来探索：
+* 首先，根据 C3v 点群的表示，写出每个键的拉曼张量。这包括：
+  * 对于 A 内竖着的键，考虑连着的两个原子和第一近邻原子，对称性为 C3v。写出此时的拉曼张量。
+  * 对于 B1 内竖着的键，它也是 C3v，它此时的拉曼张量是 h 下稍微变动的结果。写下这个结果。
+  * 对于 A 到 B1 的横着的键，它是 C3v 。写下这个结果。
+  * 对于 B1 到 C 的横着的键，它是 C3v 。写下这个结果为之前的结果的微微变动。
+  * 对于其它键，根据对称性由上面的结果直接写出。
+* 写出各个模式的拉曼张量（上面的线性组合）。即可以直接看到结果。
+*/
 
-18 of them are classified as negligible-polar phonons,
+// 我们计算了这些声子的拉曼张量与线宽，并与实验对比，结果如表所示。
-  since 
-
-// TODO: 列个表格
-
-
-// 
-// 
 // 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。
@@ -185,26 +206,26 @@ In these 18 modes, the vibrations of two Si atoms are approximately opposite to
 //   拉曼散射强度足够大且极性不强的模式，它们在拉曼散射谱上可以看到，且频率与拉曼入射光方向无关；
 //   极性声子，它们在拉曼散射谱上可以看到，不仅频率与入射光方向有关，而且可与载流子发生一些相互作用。
 
-Phonons in defect-free 4H-SiC are calculated at A-$Gamma$ and $Gamma$-M,
+// Phonons in defect-free 4H-SiC are calculated at A-$Gamma$ and $Gamma$-M,
-  as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
+//   as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
-Raman active phonons are very close to $Gamma$,
+// Raman active phonons are very close to $Gamma$,
-  as indicated by the points in the figure.
+//   as indicated by the points in the figure.
-Because of the consistency of the most phonon modes near $Gamma$,
+// Because of the consistency of the most phonon modes near $Gamma$,
-  most of the phonon frequencies are insensitive to the direction of the incident light.
+//   most of the phonon frequencies are insensitive to the direction of the incident light.
-However, some phonons have strong polarities,
+// However, some phonons have strong polarities,
-  which leads to long-range Coulomb interactions between phonons,
+//   which leads to long-range Coulomb interactions between phonons,
-  and results in different frequencies near $Gamma$,
+//   and results in different frequencies near $Gamma$,
-  as shown by the two lines in the figure.
+//   as shown by the two lines in the figure.
-Thus, we divide the phonons of defect-free 4H-SiC into three categories:
+// Thus, we divide the phonons of defect-free 4H-SiC into three categories:
-  (1) Raman inactive or too weak Raman intensity,
+//   (1) Raman inactive or too weak Raman intensity,
-    which are invisible in the Raman scattering spectrum;
+//     which are invisible in the Raman scattering spectrum;
-  (2) Raman active phonons with strong polarities,
+//   (2) Raman active phonons with strong polarities,
-    which are visible in the Raman scattering spectrum,
+//     which are visible in the Raman scattering spectrum,
-    and their frequencies are independent of the direction of the incident light;
+//     and their frequencies are independent of the direction of the incident light;
-  (3) Polar phonons,
+//   (3) Polar phonons,
-    which are visible in the Raman scattering spectrum,
+//     which are visible in the Raman scattering spectrum,
-    and their frequencies depend on the direction of the incident light,
+//     and their frequencies depend on the direction of the incident light,
-    and can interact with carriers.
+//     and can interact with carriers.
 
 #page(flipped: true)[
   #let m(n, content) = table.cell(colspan: n, content);