價電子Valence e⁻
分數Score 0 時間Time 0:00
FA引力Attraction
FR斥力Repulsion
FN淨力Net
解離巨獸Entropy Crack
AEML, NTUST
Space / 點擊 繼續 ▶Space / Tap to continue ▶
CASE FILE No. 02
真相的平衡點The Equilibrium Point
微觀破案特攻Micro-Investigation ACT
工程材料 Ch.02・原子結構與原子間鍵結Engineering Materials Ch.02 · Atomic Structure & Interatomic Bonding
「真相永遠只有一個——所有材料的命運，都決定在平衡距離 r₀ 之中！」"There is only one truth — every material's fate is decided at the equilibrium distance r₀!"
© 2026 AEML, NTUST ・ 先進工程材料實驗室・國立臺灣科技大學 營建工程系Advanced Engineering Materials Lab · Dept. of Civil & Construction Eng., NTUST 教師驗證Teacher verify
選擇特偵員Choose Your Investigator
朝霧 晴Haru Asagiri
均衡型・羽織偵探Balanced · Haori Detective
沉著冷靜的推理少年，量子衝刺距離較長。A calm young sleuth. Longer Quantum Dash.
速度Speed
跳躍Jump
衝刺Dash
櫻庭 小春Koharu Sakuraba
跳躍型・櫻巫女Jumper · Sakura Shrine Maiden
元氣滿滿的巫女偵探，跳得最高、滯空最久。A cheerful shrine-maiden sleuth. Highest, floatiest jump.
速度Speed
跳躍Jump
衝刺Dash
狐森 紺Kon Kitsunemori
疾風型・狐耳忍偵Speedster · Fox-eared Ninja Sleuth
來去如風的狐耳少年，奔跑與衝刺最快。A fox-eared whirlwind. Fastest run and dash.
速度Speed
跳躍Jump
衝刺Dash
設定Settings
背景音樂Music
音效Sound FX
主音量Master Volume
音樂音量Music Volume
音效音量FX Volume
全螢幕Full Screen
繪圖Renderer
畫質Quality
螢幕搖桿Touch Pad
劇情提示Story Hints
語言Language
按鍵Controls
自訂按鍵Customize Keys
點擊按鍵欄位後，按下新的按鍵。方向鍵永遠可用於移動。Click a key slot, then press a new key. Arrow keys always work for movement.
左移Move Left
右移Move Right
下（穿越平台）Down (drop through)
跳躍Jump
量子衝刺Quantum Dash
平衡打擊／符咒Strike / Talisman
鍵結鉤爪Bonding Grapple
切換（自旋／護盾）Switch (spin / shield)
暫停Pause
暫停Paused
存檔碼：可複製保存，換裝置時貼上即可讀檔（瀏覽器清除資料也不怕）。Save code: copy it to keep a backup or move to another device.
推理時間Deduction Time
案件陷入膠著……The case has stalled...
價電子耗盡了！從最近的檢查點重新搜查吧。You ran out of valence electrons! Resume from the last checkpoint.
結案程序：建立鈣 Ca（Z = 20）的電子檔案Case Closure: Profile Calcium Ca (Z = 20)
依遞建原理與對角線規則，依序點選副殼層證物。發光的是你在關卡中蒐集到的證物（每個 +100 分）。Using the Aufbau principle and the diagonal rule, tap the subshell evidence in order. Glowing items are evidence you collected in the cases (+100 each).
Σ e⁻ = 0 / 20
第 02 號結案檔案卡Case File No. 02 — Closed
請將檔案卡上傳至 Moodle／Zuvio 作為平時作業。認證碼可由教師驗證。Upload the card to Moodle / Zuvio as homework. The token can be verified by your instructor.
操作說明How to Play
鍵盤（預設）Keyboard (default)
AD / ←→ 移動Move
W / ↑ / K 跳躍（按住跳更高）Jump (hold for higher)
Shift / J 量子衝刺（空中也可）Quantum Dash (works in mid-air)
Space 平衡打擊／發射符咒Equilibrium Strike / throw talisman
E / L 鍵結鉤爪（再按一次放開）Bonding Grapple (press again to release)
Q / I 切換自旋 ↑↓（第二關）／護盾模式（第三關）Switch spin ↑↓ (Case 2) / shield mode (Case 3)
P / Esc 暫停Pause
微觀搜查技巧Investigation Techniques
⚛ 斥力反衝：貼近發光原子核時按跳躍，借斥力高高彈起。⚛ Repulsive Bounce: jump right next to a glowing nucleus to launch high.
⛓ 共價鉤爪：金色鍵結點可以鉤住並擺盪過深淵。⛓ Covalent Grapple: hook gold bonding nodes and swing over pits.
🌊 金屬滑行：藍色自由電子海可高速滑行。🌊 Metallic Glide: blue electron seas boost your speed.
🧲 離子磁吸：± 磁壁可吸附，貼牆跳躍可蹬牆。🧲 Ionic Cling: stick to ± walls and wall-jump.
⚖ 平衡打擊：指針到達 r₀（FN = 0）時按 Space！⚖ Equilibrium Strike: press Space when the needle reaches r₀ (FN = 0)!
❤ 生命值就是價電子（最多 8 個，八隅體）。❤ Your health is your valence electrons (max 8 — the octet).
教師驗證認證碼Verify a Token
⟳
請將裝置橫向放置以獲得最佳體驗Rotate your device to landscape for the best experience⚠ 此副本已被修改。原始作品：AEML, NTUST。認證碼功能已停用。⚠ This copy has been modified. Original work: AEML, NTUST. Tokens are disabled.
AEML, NTUST
AEML, NTUST — Engineering Materials Ch.02 Micro-Investigation ACT
案件Case白鷺 葵（導航員）Aoi Shirasagi (Navigator)解離巨獸Entropy Crack花守 鈴Suzu Hanamori朝霧 晴Haru Asagiri櫻庭 小春Koharu Sakuraba狐森 紺Kon KitsunemoriCASE 1CASE 1第一關：失衡的原子核The Missing Decimals原子序 Z・原子量・同位素・質量數Atomic number Z · atomic weight · isotopes · mass numberCASE 2CASE 2第二關：機率迷霧與量子跳躍The Electron's AlibiBohr 模型 vs 波動力學・四個量子數・包立不相容原理Bohr vs wave-mechanical model · four quantum numbers · Pauli exclusionCASE 3CASE 3第三關：極性風暴與價電子掠奪Profiling on the Periodic Table價電子・電負度差 Δχ・離子鍵 vs 共價鍵Valence electrons · electronegativity difference Δχ · ionic vs covalentCASE 4 ・ FINALCASE 4 · FINAL終極 BOSS 戰——破裂邊界Forces at WarFN = FA + FR・EN 曲線・r₀ 與 E₀・熔點・熱膨脹F_N = F_A + F_R · E_N curve · r₀ and E₀ · melting point · thermal expansion這裡是 AEML 微觀搜查本部，我是導航員白鷺葵。This is AEML Micro-Investigation HQ. I'm your navigator, Aoi Shirasagi.鋼橋檢驗報告寫著：Fe 的原子量 = 55.85。如果原子核只由「整數個」質子與中子組成，這個 .85 是從哪裡來的？The steel-bridge report says Fe's atomic weight is 55.85. If a nucleus is made of a whole number of protons and neutrons, where does the .85 come from?縮小裝置啟動！真相，就藏在原子裡。Shrink device on! The truth is hiding inside the atom.Case 1：失衡的同位素反應爐。收集 6 顆質子（Z = 6），重建碳原子核！Case 1: the unbalanced isotope reactor. Collect 6 protons (Z = 6) to rebuild a carbon nucleus!Z 決定元素身分——Z 一變，元素就換人。中子數 N 可以改變，形成同位素；質量數 A ≈ Z + N。Z decides the element — change Z and it's a different element. The neutron number N can vary, giving isotopes; mass number A ≈ Z + N.注意：每收集一顆中子，你的質量變大，重力也跟著變大！Careful: every neutron you collect adds mass — and your gravity grows with it!碳原子核重建完成！你組出的是 C-{A}（Z = 6、N = {N}）。Carbon nucleus rebuilt! You formed C-{A} (Z = 6, N = {N}).天然碳是 C-12 與 C-13 的固定比例混合。週期表上的原子量是各同位素的「加權平均」，所以帶小數點——.85 的真兇，就是同位素！Natural carbon is a fixed mix of C-12 and C-13. The atomic weight on the periodic table is a weighted average of the isotopes — so it has decimals. The culprit behind .85 is the isotope!Case 2：電子的不在場證明。Bohr 說電子沿固定圓形軌道運行——但那是被推翻的證詞。Case 2: The Electron's Alibi. Bohr said electrons ride fixed circular orbits — testimony that has been overturned.發光的圓形「軌道」一踩就碎！請站在藍色電子雲上——那是 Ψ² 機率最高的區域。The glowing circular "orbits" shatter underfoot! Stand on the blue clouds — where Ψ² is highest.軌域電梯：一個軌域最多兩顆電子，而且自旋必須相反。按「切換」改變你的自旋 ↑／↓！Orbital elevators: one orbital holds at most two electrons, with opposite spins. Press Switch to flip your spin ↑/↓!沿路收集副殼層證物（1s²、2s²……），結案時要用來建立鈣的檔案。小心假證物！Collect subshell evidence (1s², 2s², …) — you'll need it to profile calcium at the end. Beware of fakes!不在場證明成立——電子從來就不在任何一個「點」上，只有 (n, ℓ, mℓ, ms) 四個身分編號。The alibi holds — the electron was never at a single point. All it has is its ID: (n, ℓ, mℓ, ms).Case 3：週期表上的側寫。離子風暴中的元素妖會掠奪你的價電子！Case 3: Profiling on the Periodic Table. Element spirits in the ion storm will steal your valence electrons!你的裝甲核心是鈉 Na（χ = 0.9）。看敵人身上的 χ，算出電負度差 Δχ。Your armor core is sodium Na (χ = 0.9). Read each enemy's χ and compute Δχ.Δχ ≥ 1.7 → 切換為「離子鍵護盾」（電子轉移）；Δχ < 1.7 → 切換為「共價鍵鎖定」（共用電子）。按「切換」換模式，按 Space 發射符咒！Δχ ≥ 1.7 → Ionic Shield (electron transfer); Δχ < 1.7 → Covalent Lock (shared electrons). Press Switch to change mode, Space to throw a talisman!核心切換為氯 Cl（χ = 3.0）！同為鹵素的敵人現在和你電負度相近——重新計算 Δχ！Core switched to chlorine Cl (χ = 3.0)! Fellow halogens now have a similar χ to yours — recompute Δχ!動機只有一個：把最外層湊成 8 個電子——不是搶、就是送、不然就共用。Δχ 的大小直接預告鍵結型態。One motive only: complete an outer shell of 8 — take, give, or share. The size of Δχ predicts the bond type.Case 4：引力與斥力的角力。解離巨獸正企圖拉斷鋼骨大樓的晶格鍵結！Case 4: Forces at War. The Entropy Crack is trying to tear apart the steel tower's lattice bonds!看上方儀表：r > r₀ 時淨力為引力——對牠發射「引力符咒」，把鍵長拉回來！Watch the gauge: when r > r₀ the net force is attractive — throw attraction talismans to pull the bond back!r < r₀ 時斥力急遽上升——踩著斥力原子核彈起，從上方反擊牠的頭頂！When r < r₀ repulsion soars — bounce off the repulsion nuclei and strike its head from above!牠疲勞時，站到金色鍵結鎖定栓前，在 r = r₀ 的瞬間按 Space，把鍵結鎖在位能井底 E₀！When it tires, stand at the golden bond-lock bolt and press Space the instant r = r₀ to lock the bond at the bottom of the well, E₀!鍵結鎖定完成！井越深（E₀ 大）→ 熔點越高；井底越陡 → 彈性模數越大；井越不對稱 → 熱膨脹越大。Bond locked! Deeper well (larger E₀) → higher melting point; steeper well bottom → higher elastic modulus; more asymmetric well → larger thermal expansion.哈哈哈！所有的鍵都會被我拉斷——熵永遠是贏家！Ha ha ha! I'll tear every bond apart — entropy always wins!拉不斷？那就把你們全部壓碎！Can't pull you apart? Then I'll crush you together!怎……怎麼可能……力量……竟然平衡了……Im... impossible... the forces... are balanced...最後一步：用你蒐集的副殼層證物，建立鈣 Ca（Z = 20）的完整電子組態——鈣是水泥化學的主角！Final step: use your subshell evidence to build the full electron configuration of calcium Ca (Z = 20) — the star of cement chemistry!完全正確！Ca：1s² 2s² 2p⁶ 3s² 3p⁶ 4s²。價電子 2 個（4s²）——Ca 失去它們成為 Ca²⁺。真相永遠只有一個！Exactly right! Ca: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s². Two valence electrons (4s²) — lose them and Ca becomes Ca²⁺. There is only one truth!推理有破綻！提示：沿對角線規則 1s → 2s → 2p → 3s → 3p → 4s 填入，電子總數必須等於 Z = 20；ℓ 只能是 0 ~ n−1。There's a hole in your deduction! Hint: follow the diagonal rule 1s → 2s → 2p → 3s → 3p → 4s; the total must equal Z = 20, and ℓ can only be 0 to n−1.「原來這棟大樓能耐震，全靠平衡距離 r₀ 與位能井 E₀ 的深淺啊！」"So this tower survives earthquakes all thanks to the equilibrium distance r₀ and the depth of the energy well E₀!"發光的原子核會產生強大斥力。貼近時按跳躍，借斥力高高彈起！Glowing nuclei push back hard. Jump right next to one to ride the repulsion skyward!看到金色鍵結點了嗎？按「鉤爪」發射共價鍵鏈盪過去，再按一次放開！See the gold bonding node? Press Grapple to fire a covalent chain and swing — press again to let go!不穩定的軌道一踩就會崩塌，快速通過！Unstable orbits crumble the moment you land — keep moving!鍵結鎖定柱！站在柱前按 Space 開始校準，指針到 r₀（FN = 0）時再按一次！A bond-lock pillar! Stand by it and press Space to start calibrating, then press again when the needle hits r₀ (FN = 0)!量子衝刺：每次離地可衝刺一次，空中也能使用！Quantum Dash: one dash per jump — it works in mid-air!自由電子海：金屬鍵的價電子不屬於任何原子。在這裡可以高速滑行！Electron sea: metallic valence electrons belong to no single atom. Glide through at high speed!離子磁壁：陰陽離子的庫倫引力會吸住你。貼牆時按跳躍可以蹬牆跳！Ionic walls: the Coulomb attraction between ± ions holds you. Jump while clinging to wall-jump!電梯上已有一顆 ↑ 電子。先切換成 ↓ 自旋再進入！This elevator already holds a ↑ electron. Switch to ↓ spin before boarding!包立不相容！兩顆電子不能擁有完全相同的四個量子數——請切換自旋！Pauli exclusion! Two electrons can't share all four quantum numbers — flip your spin!假證物！{X} 不存在：ℓ 只能是 0 ~ n−1。Fake evidence! {X} doesn't exist: ℓ can only be 0 to n−1.取得證物 {X}Evidence found: {X}Z = {Z} → {E}：Z 一變，元素就換人！Z = {Z} → {E}: change Z, change the element!N = {N}：質量數 A = {A}，重力 ×{G}N = {N}: mass number A = {A}, gravity ×{G}鳥居未開啟：需要 Z = 6（目前 Z = {Z}）。The torii is sealed: Z = 6 required (now Z = {Z}).鳥居未開啟：碳的穩定原子核需要 N ≥ 6（目前 N = {N}）。The torii is sealed: a stable carbon nucleus needs N ≥ 6 (now N = {N}).離子鍵！Δχ = {D} → 電子轉移成功Ionic bond! Δχ = {D} → electron transferred共價鍵！Δχ = {D} → 共用電子鎖定Covalent bond! Δχ = {D} → electrons shared模式錯誤！Δχ = {D}，應使用「{M}」Wrong mode! Δχ = {D} — use {M}價電子被掠奪了！Valence electron stolen!檢查點：石燈籠已點亮Checkpoint: stone lantern lit已存檔Saved已讀檔Loaded此欄位沒有存檔This slot is empty存檔碼無效Invalid save code存檔碼已產生並複製Save code created and copied此裝置無法使用 WebGPU，已改用軟體繪圖WebGPU isn't available on this device — switched to software rendering此瀏覽器不支援原生全螢幕，已改用滿版模式Native full screen isn't supported here — using full-window mode完美平衡！PERFECTPERFECT EQUILIBRIUM!接近平衡 GOODGOOD — almost balanced太近了！斥力 FR 把你彈開Too close! Repulsion FR blasts you away太遠了！引力 FA 不足Too far! Attraction FA is too weak鍵結鎖定！通道已建立Bond locked! A path has formed受到攻擊！Ouch!鍵被拉斷了！鋼骨晶格受損The bond snapped! The steel lattice is damaged斥力反擊！Repulsive counter!引力命中！r 縮短Attraction hit! r shortens巨獸疲勞了！前往鍵結鎖定栓！The beast is exhausted! Go to the bond-lock bolt!C-12：最常見的穩定同位素（約 98.9%）C-12: the most common stable isotope (~98.9%)C-13：穩定同位素（約 1.1%）C-13: a stable isotope (~1.1%)C-14：放射性同位素，碳十四定年就是靠它！C-14: radioactive — the basis of radiocarbon dating!中子過多，原子核不穩定！Too many neutrons — the nucleus is unstable!核心切換：Cl（χ = 3.0）Core switched: Cl (χ = 3.0)全螢幕Full screen修改過的副本——無法產生有效認證碼Modified copy — no valid token can be produced請輸入姓名與學號Please enter your name and student ID質子 ZProtons Z中子 NNeutrons N質量數 AMass no. A重力Gravity自旋 msSpin ms證物Evidence核心Core模式Mode離子鍵護盾Ionic Shield共價鍵鎖定Covalent Lock鍵長 r／r₀Bond length r / r₀（無）(none)氫 HHydrogen H氦 HeHelium He鋰 LiLithium Li鈹 BeBeryllium Be硼 BBoron B碳 CCarbon C開啟ON關閉OFF目前：WebGPU 硬體繪圖（含光暈後製）Now: WebGPU hardware rendering (with bloom post-FX)目前：軟體繪圖（Canvas 2D）Now: software rendering (Canvas 2D)此裝置不支援 WebGPU，使用軟體繪圖WebGPU not supported on this device — using software請按鍵……Press a key...存檔Save讀檔Load檔案Slot（空白）(empty)推理正確！Correct deduction!推理有誤……Not quite...✔ 認證碼有效✔ Valid token✘ 認證碼無效或與學號不符✘ Invalid token or wrong student ID第 02 號結案檔案卡CASE FILE No. 02 — CLOSED真相的平衡點：微觀破案特攻The Equilibrium Point: Micro-Investigation ACT偵探Detective學號Student ID特偵員Investigator總分Score答題正確率Quiz accuracy平衡打擊Equilibrium strikes平均時機誤差Avg. timing error破案時間Clear time評等Rank破案認證碼Verification token鈣的電子組態Ca configuration工程材料 Ch.02 原子結構與原子間鍵結Engineering Materials Ch.02 Atomic Structure & Interatomic Bonding真相之眼Eye of Truth名偵探Master Detective見習偵探Junior Detective搜查新人Rookie Investigator
氯有兩種同位素：³⁵Cl（34.969 amu，75.77%）與 ³⁷Cl（36.966 amu，24.23%）。Cl 的平均原子量約為？Chlorine has two isotopes: ³⁵Cl (34.969 amu, 75.77%) and ³⁷Cl (36.966 amu, 24.23%). Its average atomic weight is about?35.00 amu35.00 amu35.45 amu35.45 amu35.97 amu35.97 amu36.00 amu36.00 amuĀ = Σ fiAi = 0.7577(34.969) + 0.2423(36.966) = 35.45 amu。原子量是加權平均，不是簡單平均。Ā = Σ fiAi = 0.7577(34.969) + 0.2423(36.966) = 35.45 amu — a weighted, not simple, average.
為什麼週期表上 Fe 的原子量是 55.85，而不是整數？Why is iron's atomic weight 55.85 instead of a whole number?實驗室秤量誤差造成Laboratory weighing error電子的質量加上去造成The electrons' mass adds the decimals量到的是各同位素依存在比例的加權平均It is a weighted average over the naturally occurring isotopes中子會隨時間衰變Neutrons decay over time我們量到的從來不是「一顆」原子，而是一整群不同同位素的混合，原子量 Ā = Σ fiAi。We never weigh "one" atom — we weigh a mixture of isotopes, so Ā = Σ fiAi.
鈰 Ce 的同位素：¹⁴⁰Ce 佔 88.45%、¹⁴²Ce 佔 11.11%，其餘極少。不用計算機，Ce 的原子量最接近？Cerium: ¹⁴⁰Ce is 88.45% and ¹⁴²Ce is 11.11%, the rest tiny. Without a calculator, Ce's atomic weight is closest to?137.9137.9139.0139.0140.1140.1141.9141.9答案一定緊貼比例最大的 ¹⁴⁰Ce（139.905），再被 ¹⁴²Ce 往上拉一點 → 約 140.12 amu。It must hug the dominant ¹⁴⁰Ce (139.905), pulled up slightly by ¹⁴²Ce → about 140.12 amu.
下列哪一組量子數 (n, ℓ, mℓ, ms) 不可能存在？Which set of quantum numbers (n, ℓ, mℓ, ms) is impossible?(2, 1, 0, +½)(2, 1, 0, +½)(3, 2, −1, +½)(3, 2, −1, +½)(2, 2, 1, −½)(2, 2, 1, −½)(1, 0, 0, −½)(1, 0, 0, −½)ℓ 只能是 0 ~ n−1。n = 2 時 ℓ 最大為 1，所以 ℓ = 2（2d）不存在。ℓ ranges from 0 to n−1. For n = 2, ℓ ≤ 1, so ℓ = 2 (a "2d") cannot exist.
(3, 2, −1, +½) 描述的電子位於哪一個副殼層？The electron (3, 2, −1, +½) is in which subshell?3s3s3p3p3d3d4d4dn = 3 → 第 3 殼層（M）；ℓ = 2 → d。mℓ = −1 是 5 個 d 軌域之一，ms = +½ 為自旋向上。n = 3 → shell M; ℓ = 2 → d. mℓ = −1 is one of five d orbitals; ms = +½ is spin-up.
第 n = 3 殼層最多可以容納幾個電子？How many electrons can the n = 3 shell hold at most?8899181832322n² = 2 × 3² = 18：3s (2) + 3p (6) + 3d (10) = 18。根源是包立不相容原理：每個軌域最多 2 個自旋相反的電子。2n² = 2 × 3² = 18: 3s (2) + 3p (6) + 3d (10). The root cause is Pauli exclusion: two opposite-spin electrons per orbital.
MgO：χMg = 1.2、χO = 3.5。依 %離子性 = {1 − exp[−(Δχ)²/4]} × 100%，約為？MgO: χMg = 1.2, χO = 3.5. Using %IC = {1 − exp[−(Δχ)²/4]} × 100%, the ionic character is about?12%12%51%51%73%73%95%95%Δχ = 2.3；(2.3)²/4 = 1.32；exp(−1.32) = 0.267 → (1 − 0.267) × 100% ≈ 73%。Δχ = 2.3; (2.3)²/4 = 1.32; exp(−1.32) = 0.267 → (1 − 0.267) × 100% ≈ 73%.
Na（…3s¹）與 Cl（…3s² 3p⁵）相遇時，最可能發生什麼事？When Na (…3s¹) meets Cl (…3s² 3p⁵), what most likely happens?兩者各拿出一個電子共用，形成共價鍵Each shares one electron — a covalent bondNa 把 3s 電子交給 Cl，形成 Na⁺ 與 Cl⁻ 的離子鍵Na gives its 3s electron to Cl, forming an ionic bond between Na⁺ and Cl⁻Cl 把電子交給 NaCl gives an electron to Na形成電子海的金屬鍵They form a metallic electron seaΔχ = 3.0 − 0.9 = 2.1（離子性約 67%）。Na 送出 1 個、Cl 收下 1 個，雙方都達到惰性氣體組態。Δχ = 3.0 − 0.9 = 2.1 (~67% ionic). Na gives one, Cl takes one — both reach a noble-gas configuration.
為什麼混凝土、陶瓷等離子性材料又硬又脆？Why are ionic materials such as ceramics and concrete hard but brittle?因為含有很多自由電子They contain many free electrons因為鍵結具有方向性且很弱Their bonds are directional and weak原子面滑移後同性離子相鄰，庫倫斥力使晶體直接斷裂After slip, like-charged ions become neighbours and Coulomb repulsion splits the crystal因為密度太低Their density is too low離子晶體只要滑移半個晶格，原本相鄰的異性離子就變成同性相鄰——滑移 = 斷裂，所以幾乎沒有塑性變形。這正是混凝土需要鋼筋的原因。Slip by half a lattice spacing puts like charges side by side — slip means fracture, so there's almost no plastic deformation. That's why concrete needs rebar.
兩原子位於平衡間距 r₀ 時，下列何者正確？At the equilibrium spacing r₀, which statement is correct?FA 與 FR 都等於零FA and FR are both zeroFN = FA + FR = 0，且位能 EN 最低（= E₀）FN = FA + FR = 0 and the energy EN is minimum (= E₀)位能 EN = 0The energy EN = 0斥力 FR 達到最大Repulsion FR is at its maximumF = dE/dr：淨力為零之處正是 E–r 曲線的最低點。FA 與 FR 並不為零，而是大小相等、方向相反。F = dE/dr: zero net force is exactly the minimum of the E–r curve. FA and FR are not zero — they are equal and opposite.
鍵結能 E₀ 越大（位能井越深）的材料，通常會怎樣？A material with a larger bonding energy E₀ (deeper well) usually has…?熔點越低a lower melting point熔點越高a higher melting point熱膨脹係數越大a larger thermal expansion coefficient彈性模數越小a smaller elastic modulus井越深，要把原子搖散所需的熱能越多 → 熔點 Tm 越高。The deeper the well, the more thermal energy is needed to shake atoms apart → higher Tm.
鑽石（強共價鍵）與鉛（弱金屬鍵）相比，下列何者正確？Comparing diamond (strong covalent) with lead (weak metallic), which is correct?鑽石：井淺而平 → Tm 低、α 大Diamond: shallow, flat well → low Tm, large α鉛：井深而陡 → E 極大Lead: deep, steep well → huge E兩者的 E–r 曲線完全相同Both have identical E–r curves鑽石：井深而陡 → Tm 高、E 極大、α 極小Diamond: deep, steep well → high Tm, huge E, tiny α同一口井解釋三個性質：井深 → 熔點、井底曲率 → 彈性模數、井的不對稱 → 熱膨脹。One well explains three properties: depth → melting point, curvature → elastic modulus, asymmetry → thermal expansion.