四极磁铁磁透镜聚焦动画（修正版）

Description

A short 3‑D animation that visualises the focusing effect of a quadrupole magnetic lens on an electron beam, exactly matching the coordinate system of the original Figure 3.

• Axes –
  • z‑axis (central axis): horizontal, pointing from right to left (negative direction), blue arrow.
  • x‑axis: horizontal to the right, red arrow.
  • y‑axis: vertical upward, green arrow.
  • Origin O is placed slightly to the right of the screen centre.
• Camera – a slanted‑top‑down view that renders the z‑axis as a horizontal line, reproducing the perspective of the source figure.
• Magnetic lens – a semi‑transparent blue rectangular prism spanning $z=0$ to $z=-L_{0}$ and labelled “磁透镜”.
• Magnetic field – cyan (#00ffff) dashed‑arrow grid:
  • $B_{x}$ points right for $y>0$ and left for $y<0$.
  • $B_{y}$ points up for $x>0$ and down for $x<0$.
• Electron – red sphere (#e74c3c) with a faint white trailing glow.
• Trajectory – orange solid line (#f39c12).
• Focal point – bright yellow dot labelled $(0,0,-f_{x})$.
• Formulas – $\displaystyle \omega^{2}= \frac{e\,v_{z}\,k_{0}}{m}$ and $\displaystyle f_{x}=L_{0}+ \frac{m\,v_{z}}{e\,k_{0}\,L_{0}}$ appear in the top‑right corner in white LaTeX.

The animation proceeds through five phases, totalling 8.5 s.

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Phases

#	Phase Name	Duration	Description
1	坐标系与磁透镜	1 s	Axes (x, y, z) and the semi‑transparent blue lens volume appear; the origin O is highlighted.
2	磁场线显示	0.5 s	Cyan dashed‑arrow grid populates the lens region, illustrating the $B_{x}$ and $B_{y}$ patterns described above.
3	入射与磁场内运动	2 s	An electron starts at $(x_{0},0,0)$ and moves negative‑z with constant speed $v_{z}$ ( $z(t)=-v_{z}t$ ). Its tiny $x$ displacement follows the linear approximation of the harmonic motion (since $\omega t\ll1$). The orange trajectory is drawn, and a white tail follows the red sphere.
4	出磁场后直线运动	2 s	Upon exiting at $z=-L_{0}$ the electron acquires a transverse velocity $v_{x}= \dfrac{e\,k_{0}\,L_{0}\,x_{0}}{m}$ and travels straight to the focal point $(0,0,-f_{x})$. The focal point appears as a yellow highlight; the straight‑line segment is drawn in orange.
5	多电子聚焦演示	3 s	Three additional electrons start from $x_{0}$, $0.8x_{0}$ and $0.6x_{0}$. Each follows the same inside‑lens motion and exits with the corresponding $v_{x}$, all converging simultaneously at the same focal point, emphasizing the lens’s focusing property.

All phase transitions are smooth cross‑fades of ~0.2 s.

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Layout

┌─────────────────────────────────────────────┐
│               TOP AREA (标题)               │
├───────────────────────┬─────────────────────┤
│                       │                     │
│   LEFT AREA (3D 场景) │   RIGHT AREA (公式) │
│                       │                     │
├───────────────────────┴─────────────────────┤
│               BOTTOM AREA (阶段标题)        │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Optional overall title (e.g., “四极磁铁磁透镜聚焦”)	Fades in with Phase 1
Left	Full 3‑D scene: axes (x red, y green, z blue), lens volume, magnetic‑field grid, electrons, trajectories, focal point	Camera starts at a slanted‑top‑down angle that makes the z‑axis appear horizontal; it gently orbits (≈30° pan) over the whole 8.5 s
Right	Formulas $\omega^{2}= \frac{e v_{z} k_{0}}{m}$ and $f_{x}=L_{0}+ \frac{m v_{z}}{e k_{0} L_{0}}$ in white LaTeX	Fixed at the top‑right corner of the screen; appears at the start of Phase 3 and stays visible
Bottom	Phase titles (“1. 坐标系与磁透镜”, “2. 磁场线显示”, …)	Updated at the beginning of each phase, fades in/out

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Notes

• Background: deep‑gray #2c3e50.
• Axes: 3‑D orthogonal arrows; lengths and tick marks visible.
• Lens volume: semi‑transparent blue rectangular prism, labelled “磁透镜”.
• Magnetic field lines: cyan dashed arrows forming a grid that follows the sign rules for $B_{x}$ and $B_{y}$.
• Electron: red sphere radius 0.1, with a faint white trailing glow.
• Trajectory: orange solid line left behind as the electron moves.
• Focal point: bright yellow dot, labelled $(0,0,-f_{x})$.
• Formulas: rendered in LaTeX, white colour, positioned top‑right.
• Camera motion: starts with the slanted‑top‑down view described above; over the 8.5 s total the camera performs a gentle 30° pan around the scene to give depth perception.
• Timing: total 8.5 s at 30 fps (≈255 frames). Phase durations are as listed; transitions are ~0.2 s cross‑fades.
• No extra text beyond the specified labels, axis names, and formulas; all information is conveyed visually.