AEML, NTUST · Advanced Engineering Materials Lab, National Taiwan University of Science and Technology
中文
中文 / English
English
命中條件:|2θaim − 2arcsin(λ/2d)| ≤ Δtol。算錯角度,子彈只會穿透。Hit rule: |2θaim − 2arcsin(λ/2d)| ≤ Δtol. A wrong angle just passes through.
λ ≥ 2d 時 sinθ > 1:這個彈種對該晶面完全無效。If λ ≥ 2d then sinθ > 1: that tube cannot reach the plane.
F = 0 的晶面完全免疫。消失的峰本身就是證據。Planes with F = 0 are immune. A missing peak is evidence.
沒裝濾片時,每一槍都附帶 Kβ 幽靈彈,在較小 2θ 產生假峰。Without a filter every shot carries a Kβ ghost at lower 2θ.
精度與 √N 成正比:精度要加倍,蓄力時間要四倍。Precision ∝ √N: double the precision costs four times the counting.
每一根峰都必須被指派。未被指派的峰,在你證明它是雜訊之前,都不算雜訊。Assign every peak. An unassigned peak is not noise until you prove it.
葛諾博士Dr. Gonio
米勒Miller
鏡影Mirage
夜鷹 λNighthawk λ
霜月Shizuki
檔案官Archivist
熟料巨像Clinker Colossus
旁白Narrator
晶Akira
光Hikari
結衣Yui
蓮Ren
空Sora
澪Mio
特長:移動速度 +15%Perk: move speed +15%
特長:體力 120Perk: 120 HP
特長:每案多 1 顆鹼液彈Perk: +1 alkali grenade
特長:蓄力加快Perk: faster charging
特長:光管散熱 +35%Perk: tube cooling +35%
特長:傷害 +10%Perk: damage +10%
玩家Human
電腦隊友CPU
遊戲手把Gamepad
線上玩家Online
光源手Source
光路手Optics
製樣手Prep
分析手Analyst
有序還是混亂Order or Chaos?
藍圖The Blueprint
座標密碼The Coordinate Cipher
對稱的不在場證明The Symmetry Alibi
看不見的目擊者The Invisible Witness
強度的指紋Fingerprint of Intensity
佈置偵訊室The Interrogation Room
不可靠的證詞The Unreliable Witness
指紋資料庫The Fingerprint Database
這是誰的陰影Whose Shadow?
被動過手腳的證物The Tampered Evidence
水泥指認會The Cement Line-up
石英兵Quartz
方解石Calcite
霰石Aragonite
氫氧化鈣 CHPortlandite CH
鈣礬石Ettringite
二水石膏Gypsum
半水石膏Hemihydrate
無水石膏Anhydrite
K₂SO₄(arcanite)Arcanite K₂SO₄
方鎂石 MgOPericlase MgO
鋁酸鹽 C₃AAluminate C₃A
鐵鋁酸鹽 C₄AFFerrite C₄AF
Alite C₃SAlite C₃S
Belite C₂SBelite C₂S
游離石灰Free lime
氫氧化鈣 CHPortlandite CH
磁鐵礦Magnetite
α-Feα-Fe
銅 CuCopper
複製兵(簡單 P)Copy (primitive P)
高指標小靶 d = 0.070 nmHigh-index target d = 0.070 nm
雙生兵 ATwin A
雙生兵 BTwin B
重晶石 BaSO₄(含鋇)Barite BaSO₄ (barium)
方矽石Cristobalite
石英標準靶Quartz standard
石英靶 (101)Quartz target (101)
玻璃相小嘍囉Glass-phase minion
鐵衛Iron Guardian
銅衛Copper Guardian
訓練靶(石英)Training target (quartz)
高鐵爐石兵Iron-rich slag soldier
石英標準靶Quartz standard
氫氧化鈣盾兵Portlandite shield
十九峰合成體Nineteen-Peak Chimera
50% 石英兵50% quartz
石膏試射靶Gypsum test target
熟料巨像Clinker Colossus
玻璃相首腦Glass Phase Leader
代碼兵Code-bot
鏡影分身Mirage clone
Kβ 假峰Kβ ghost
立方 PCubic P
立方 I(體心)Cubic I
立方 F(面心)Cubic F
斜方 C(底心)Orthorhombic C
正方 ITetragonal I
六方 PHexagonal P
立方 CubicCubic
正方 TetragonalTetragonal
斜方 OrthorhombicOrthorhombic
六方 HexagonalHexagonal
菱方 RhombohedralRhombohedral
單斜 MonoclinicMonoclinic
三斜 TriclinicTriclinic
簡單 P:全部 (hkl) 出現Primitive P: all (hkl) present
體心 I:h + k + l 為偶數Body-centred I: h + k + l even
面心 F:h、k、l 全奇或全偶Face-centred F: h, k, l all odd or all even
底心 C:h + k 為偶數Base-centred C: h + k even
有峰peak
無峰(F = 0)no peak (F = 0)
未測untested
鹼液閥門Alkali valve
晶胞生成點Cell spawner
結界 1Barrier 1
結界 2Barrier 2
結界 3(方向)Barrier 3 (direction)
結界 4Barrier 4
結界 5(六方)Barrier 5 (hexagonal)
鏡台:判斷對稱Mirror stand: symmetry
試樣台Specimen stand
晶格判定終端機Lattice terminal
校正台Calibration desk
工作台:側向裝填Bench: side-load
資料庫終端機Database terminal
掃描台Scan desk
回報終端機Report terminal
內標信標Internal standard
同學的彈藥Classmate's ammo
製樣工作台Prep bench
製樣工作台:製作彈藥Prep bench: make ammo
研磨方式Grinding
低速手動研缽Hand mortar (gentle)
環磨機Ring mill
研磨時間Time
2 分鐘2 min
5 分鐘5 min
10 分鐘10 min
30 分鐘30 min
液體Liquid
乾磨Dry
加乙醇With ethanol
篩分Sieve
不過篩No sieve
#325 篩(45 μm)#325 sieve (45 μm)
裝填方式Loading
正面壓實Front-pressed
側向裝填Side-loaded
製作彈藥Make ammo
>45 μm 未過篩>45 μm unsieved
5–45 μm 偏粗5–45 μm coarse
0.2–5 μm 理想0.2–5 μm ideal
<0.2 μm 過細<0.2 μm too fine
過度研磨(晶格應變)Over-ground (strain)
標準彈藥Standard ammo
尚未製作Not made yet
氧化鈹煙霧(弱吸收)BeO smoke (weak absorber)
方矽石煙霧(吸收相同)Cristobalite smoke (same absorber)
氯化鉀煙霧(強吸收)KCl smoke (strong absorber)
含鐵重相煙霧(強吸收)Heavy iron-phase smoke (strong)
調查Use
目標:Objective:
提示:試試 2θ ≈ {tt}Hint: try 2θ ≈ {tt}
提示:這個彈種 λ ≥ 2d,換一種波長更短的靶材。Hint: λ ≥ 2d for this tube; switch to a shorter wavelength.
2θ = 2·arcsin(λ / 2d)2θ = 2 arcsin(λ / 2d)
沒有目標No target
限 {t} 彈{t} only
19 根峰,身分不明。到掃描台記錄峰位。19 unknown peaks. Record them at the scan desk.
30–35° 擠滿重疊的峰……30–35° crowded with overlapping peaks...
(不需計算角度)(no angle needed)
練習用自動瞄準中Training auto-aim is on
校準檔鎖定calibration lock
側向裝填side-loaded
螢光霧fluorescence
彈藥Ammo
鹼液Alkali
灼燒Ignite
蓄力未解鎖charge locked
寬鼓包Broad hump
峰重疊,無法歸屬Overlap: cannot attribute
被矽酸鹽淹沒Drowned by silicates
此靶不收這個彈種Wrong tube for this target
護盾Shield
幻影Illusion
已記錄Recorded
過熱停機Overheated
這個彈種用完了,換回 Cu。Out of this ammo; switch back to Cu.
{p} 倒下了!4 秒後復活。{p} is down! Back in 4 s.
這項設定由「{r}」負責。That setting belongs to the {r}.
尚未解鎖Not unlocked yet
狹縫被鎖在校準檔!The slit is locked in calibration!
附近沒有可調查的東西Nothing to use here
沒有手榴彈了No grenades left
鹼液溶解了 {n} 個玻璃相!The alkali dissolved {n} glass minion(s)!
沒有玻璃相被溶解(結晶不受鹼液影響)。No glass dissolved (crystals are unaffected).
這裡用不上這種萃取。That extraction does nothing here.
冷卻中:{n} 秒Cooling down: {n} s
旋轉試樣中Spinning sample
標記峰位!Peak here!
需要濾片!Need a filter!
彈藥偏粗!Ammo too coarse!
推理時間Deduction
隊友推理中……A teammate is deducing...
{p} 正在解謎,時間暫停。{p} is solving; time is frozen.
不對喔,再想一次。Not quite; think again.
送出Submit
關閉Close
下一題Next
結案!Close the case!
下一案Next case
重打本案Replay
匯出學習紀錄Export log
回主選單Main menu
存檔Save
讀檔Load
刪除Delete
請按新按鍵…Press a key...
偵探隨堂考Detective Quiz
隨堂考完成Quiz complete
答對了!Correct!
差一點。Not quite.
等待房主結案…Waiting for the host...
結案CASE CLOSED
真相大白The Truth
準確度Accuracy
推理分Reasoning
效率Efficiency
隨堂考quiz
推理一次答對first-try deductions
已結案closed
已加入joined
等待加入open seat
自動(依場景)Auto (by scene)
自動存檔Autosave
存檔欄 {n}Slot {n}
(空)(empty)
沒有可用的存檔No usable save
已存到欄位 {n}Saved to slot {n}
存檔失敗(瀏覽器空間不足或封鎖)Save failed (storage full or blocked)
讀檔完成Loaded
還沒有學習紀錄No log yet
只有房主可以重新開始Only the host can restart
確定要重新開始本案嗎?Restart this case?
本案重新開始Case restarted
已自動存檔,可從「繼續」回來。Autosaved; use Continue to come back.
WebGPU 啟動中…Starting WebGPU...
WebGPU 硬體加速中WebGPU hardware active
此裝置不支援 WebGPU,已改用軟體繪圖WebGPU unavailable; using software
軟體繪圖Software rendering
房號應為 6 碼英數字Room code must be 6 letters/digits
連線中…Connecting...
無法連上中繼,稍後自動重試Relay unreachable; retrying
房號 {n}:等待玩家加入Room {n}: waiting for players
已連上中繼,等待房主…Connected; waiting for the host...
房主離線了The host went offline
連線中斷,重新連線中…Connection lost; reconnecting...
已加入房間,等待房主出發Joined; waiting for the host to start
房間已滿,或房主沒有開放「線上玩家」座位Room full, or no Online seat open
{p} 加入了!{p} joined!
{p} 離線,暫時由系統代管{p} disconnected
{p} 回來了!{p} is back!
已複製連結Link copied
截距Intercepts
原點已移到 O′origin moved to O′
箭頭終點座標Arrow tip
俯視陣形Formation (top view)
↑ 正立分身↑ upright clone
↓ 倒立分身(較低處)↓ upside-down clone (lower)
轉 {n}° 重合repeats every {n}°
標準值 2θCertified 2θ
量測值 2θObserved 2θ
峰高讀數Peak reading
巨像的峰形Colossus profile
參考reference
示意圖,非實測schematic, not measured
同學的筆記Classmate's notes
環磨機・乾磨・30 分鐘「確保夠細」Ring mill, dry, 30 min "to be sure"
圖譜:峰變寬、峰形歪斜、多了一根小峰?Pattern: broad, lopsided peaks, and a new small one?
可用窗口 0.2–5 μmusable window 0.2–5 μm
峰寬化broadening
微吸收・消光・晶粒太少microabsorption, extinction, too few grains
資料庫終端機(PDF search–match)Database terminal (PDF search–match)
已記錄的峰:勾選最強兩根Recorded peaks: tick the two strongest
Kβ?Kβ?
化學篩選Chemistry filter
搜尋Search
勾選最強兩根峰後搜尋(容許 ±0.004 nm)。化學篩選只保留 OPC + 水 + 砂可能含有的元素:Ca Si Al Fe Mg S K Na O H C。比對分數只是參考。Tick the two strongest peaks and search (±0.004 nm). The chemistry filter keeps only elements OPC + water + sand can contain: Ca Si Al Fe Mg S K Na O H C. Scores are only a guide.
沒有符合的卡片No matching cards
品質quality
卡片Card
品質評等:{q} 年份:{y}(* = 資料完整可靠;遊戲用卡號)Quality: {q}, year {y} (* = well characterised; game card IDs)
接受此相並扣除其峰Accept and subtract its peaks
上:你的圖譜 下:卡片top: your pattern, bottom: the card
PopPop
和風Japanese
JazzJazz
節奏明快Up-tempo
Lo-fiLo-fi
RockRock
FunkFunk
8-bit8-bit
Boss 戰Boss
布拉格偵探社Bragg Detective Agency
櫻與晶格Sakura and Lattices
簡報室搖擺Briefing Room Swing
出發!Off We Go!
午後的測角器Afternoon Goniometer
米勒指標藍調Miller Index Blues
單位晶胞散步Unit Cell Stroll
都節的鏡廳Hall of Mirrors
布拉格衝刺Bragg Dash
消光搖滾Extinction Rock
Kα 流行Kα Pop
索勒放克Soller Funk
高角度追擊High-Angle Chase
8-bit 繞射8-bit Diffraction
陽音階狂奔Yo-Scale Sprint
推理爵士Deduction Jazz
散射因子夜曲Scattering Factor Nocturne
搜尋比對藍調Search-Match Blues
巨像之怒Wrath of the Colossus
玻璃相的陰謀Glass Phase Conspiracy
熟料窯決戰Showdown at the Kiln
真相大白The Truth Revealed
雨中的實驗室Rainy Lab
結案之歌Case Closed
前進Forward
後退Back
左移Left
右移Right
射擊(按住蓄力)Fire (hold to charge)
2θ +0.05°2θ +0.05°
2θ −0.05°2θ −0.05°
2θ +1°2θ +1°
2θ −1°2θ −1°
輸入 2θType 2θ
換目標Next target
調查Use
投擲Throw
旋轉試樣Spin sample
換靶材Next tube
換接收狹縫Next slit
索勒狹縫Soller
β 濾片β filter
單光儀Monochromator
標記:峰位Ping: peak
標記:需要濾片Ping: filter
標記:彈藥偏粗Ping: coarse ammo
暫停Pause
設定 / 返回Settings / back
全螢幕Full screen
布拉格偵探社,實驗室。證物袋裡有一塊預鑄板碎片與一顆飛灰玻璃珠:同樣灰色、同樣是矽酸鹽、手感也一樣。Bragg Detective Agency, the lab. An evidence bag holds a precast-panel fragment and a fly-ash glass bead: both grey, both silicate, both feel like stone.
新人,歡迎!我是葛諾博士,偵探社的發明家。怪事來了——證物活過來了,而且外觀一模一樣。Welcome, rookie! I'm Dr. Gonio, the agency's inventor. Strange business: the evidence has come alive, and they all look the same.
一個在鹼液中幾天就崩解,另一個卻能撐幾十年。化學成分相同,命運卻相反——差別不在「原子是什麼」,而在「原子怎麼排」。One dissolves in alkali within days, the other survives decades. Same chemistry, opposite fate: the difference is not WHAT the atoms are, but HOW they are arranged.
這把是練習用繞射槍,已開啟自動瞄準。點一下嫌犯(或按 Tab)鎖定,再按「射擊」。結晶會回應一根尖峰——仔細看哪些不會。Here's a training diffraction gun with auto-aim. Tap a suspect (or press Tab) to lock it, then FIRE. A crystal answers with a sharp peak; watch which ones don't.
嗯?它只冒出一個寬鼓包,毫髮無傷!沒有長程有序,X 光就沒有可干涉的對象——這是玻璃相的小嘍囉。Hm? Only a broad hump, and not a scratch! Without long-range order the X-rays have nothing to interfere with. That's a glass-phase minion.
繞射槍對它無效。左邊牆上有鹼液閥門,走過去按「調查」。飛灰與爐石的玻璃相,正因為是非晶質才具有反應性。The gun can't touch it. There's an alkali valve on the left wall: walk over and press Use. Fly-ash and slag glass are reactive precisely because they are amorphous.
鹼液手榴彈裝填完畢!對玻璃相按「投擲」——它們會在鹼液中溶解,結晶兵則不受影響。Alkali grenades loaded! Throw them at the glass phase: it dissolves in alkali, while the crystalline soldiers are unaffected.
漂亮!證物室清空了。Splendid! The evidence room is clear.
長程、重複的三維有序產生可標定的尖銳繞射峰;非晶只有短程有序,只留下寬鼓包。外觀習性只暗示結構,唯有繞射能證明它。Long-range, repeating 3-D order gives sharp, indexable peaks; amorphous matter has only short-range order and leaves a broad hump. Habit only hints at structure; diffraction proves it.
擊破證物兵。哪些打不動?Defeat the evidence soldiers. Which ones won't take damage?
自動瞄準已開啟:鎖定後直接射擊,看出現的是「尖峰」還是「寬鼓包」。Auto-aim is on: lock and fire, and see whether a sharp peak or a broad hump appears.
用鹼液手榴彈(投擲)溶解所有玻璃相,並擊破結晶兵。Dissolve every glass minion with alkali grenades (Throw) and defeat the crystals.
真相永遠只有一個——差別不在原子種類,而在排列的週期性。There is only one truth: the difference is not the atoms, it is the periodicity of their arrangement.
預鑄板碎片與飛灰玻璃珠化學成分相近,命運卻相反。根本差異是什麼?A precast fragment and a fly-ash bead have similar chemistry but opposite fates. What is the root difference?
- 原子排列是否具長程有序Whether the atoms have long-range order
- 所含元素種類不同Different elements
- 顏色深淺不同Different colour
- 密度不同Different density
差別在排列,不在原子:結晶具長程週期性,玻璃沒有。Arrangement, not atoms: crystals are periodic over long range, glass is not.
非晶質(玻璃)的 XRD 圖譜長什麼樣?What does an amorphous (glass) XRD pattern look like?
- 一組尖銳峰A set of sharp peaks
- 完全沒有任何訊號No signal at all
- 一個寬鼓包,沒有尖峰One broad hump, no sharp peaks
- 只有 Kβ 峰Only Kβ peaks
只有短程有序,散射無法在特定角度建設性干涉,只剩寬鼓包(broad hump)。Only short-range order: no sharp constructive interference, just a broad hump.
飛灰與爐石的玻璃相為何具有反應性?Why are fly-ash and slag glasses reactive?
- 因為它們含有特別多的鐵They contain extra iron
- 因為它們是非晶質,沒有長程有序They are amorphous, with no long-range order
- 因為它們顆粒特別粗Their particles are coarse
- 因為它們是單晶They are single crystals
講義:水泥領域同理,飛灰與爐石的玻璃相正因非晶才具反應性。From the slides: their glass is reactive precisely because it is amorphous.
晶體習性(habit)與晶體結構的關係是?How does crystal habit relate to crystal structure?
- 兩者完全相同They are identical
- 兩者毫無關係They are unrelated
- 習性決定結構Habit determines structure
- 習性受結構控制,但不等同於結構Habit is controlled by structure but not equivalent to it
生長條件(速率、雜質、空間)可改變習性而不改變結構。習性是外部證據,結構才是內部真相。Growth conditions change habit without changing structure.
歡迎來到我的無限倉庫!敵人正從單位晶胞一個接一個複製出來。Welcome to my Infinite Warehouse! Enemies keep copying themselves out of unit cells.
孩子,別去畫原子——去畫那個「把原子放上去的規則」。晶格點不是原子,而是「地址」:你在一個地址放了什麼,其他每個地址就都放著同樣的東西。Don't draw the atoms, my boy: draw the RULE that puts them there. Lattice points are addresses, not atoms: whatever sits at one address sits at every other.
六個晶格參數 a、b、c、α、β、γ 就是整張藍圖。每個生成點都寫著參數:走過去「調查」,選對七大晶系之一,就能關閉它。Six parameters a, b, c, α, β, γ are the whole blueprint. Each spawner shows its parameters: walk up, Use it, and pick the right crystal system to shut it.
生成點全部關閉!不過……十四種布拉維格子正排隊登場。All spawners closed! But... the fourteen Bravais lattices are lining up.
規則:先說出它「每個晶胞含幾個晶格點」,護盾才會解除。P 型 1 點、I 型 2 點、F 型 4 點——有幾點,就要連續命中幾次!The rule: name how many lattice points the cell contains and its shield drops. P has 1, I has 2, F has 4: hit it once per point!
七種盒子就足以裝下宇宙中所有晶體;十四名嫌犯,就是全部的指認名單。Seven boxes hold every crystal in the universe; fourteen suspects make the whole line-up.
但繞射儀從來看不到盒子——它看到的永遠是「平面」。下一案,我們要替平面命名。But a diffractometer never sees a box, only PLANES. Next case, we learn to name them.
調查三個生成點,依 a、b、c、α、β、γ 判斷晶系並關閉它們;清除複製兵。Use the three spawners, identify each crystal system from a, b, c, α, β, γ, and close them; clear the copies.
立方 a=b=c、全 90°;正方 a=b≠c;斜方 a≠b≠c、全 90°;六方 a=b≠c、γ=120°;菱方 a=b=c、α=β=γ≠90°;單斜 β≠90°;三斜 無限制。Cubic a=b=c, all 90°; tetragonal a=b≠c; orthorhombic a≠b≠c, all 90°; hexagonal a=b≠c, γ=120°; rhombohedral a=b=c, α=β=γ≠90°; monoclinic β≠90°; triclinic none.
布拉維遊行:答出晶格點數解除護盾,再依點數連續命中。Bravais parade: state the lattice-point count to drop the shield, then hit once per point.
角落 8 × 1/8 = 1;體心 +1;六個面心 6 × 1/2 = 3;兩個底心 2 × 1/2 = 1。Corners 8 × 1/8 = 1; body centre +1; six face centres 6 × 1/2 = 3; two base centres 2 × 1/2 = 1.
這個生成點的晶胞參數如圖。它屬於哪一個晶系?This spawner's cell parameters are shown. Which crystal system is it?
這個晶胞(白點在角落、金點在心化位置)共含幾個晶格點?How many lattice points does this cell contain (white at corners, gold at centring sites)?
生成點關閉!Spawner closed!
判斷錯誤,生成點反而吐出更多複製兵!Wrong system: the spawner spits out more copies!
護盾解除!需要命中 {n} 次。Shield down! {n} hits needed.
點數錯了,晶胞反擊!Wrong count: the cell strikes back!
真相永遠只有一個——無限大的晶體,只需要一個小晶胞加上一條重複規則就能完整描述。There is only one truth: an infinite crystal is fully described by one small cell and one repetition rule.
面心立方(F)晶胞含幾個晶格點?How many lattice points are in a face-centred cubic (F) cell?
- 1
- 2
- 4
- 8
8 × 1/8 + 6 × 1/2 = 4。8 × 1/8 + 6 × 1/2 = 4.
哪一組參數描述六方晶系?Which parameters describe the hexagonal system?
- a = b ≠ c, α = β = 90°, γ = 120°
- a = b = c, α = β = γ = 90°
- a ≠ b ≠ c, α = β = γ = 90°
- a = b = c, α = β = γ ≠ 90°
其餘分別為立方、斜方、菱方。The others are cubic, orthorhombic and rhombohedral.
為何常使用較大的非原始晶胞?Why use a larger, non-primitive cell at all?
- 因為原始晶胞不存在Primitive cells don't exist
- 因為它能讓對稱性一目了然It makes the symmetry obvious
- 因為它含有較多原子It contains more atoms
- 因為比較省筆墨It saves ink
最小不代表最清楚:偵探挑「最能顯示規律」的描述方式。Smallest is not clearest; pick the description that reveals the pattern.
晶格點代表什麼?What is a lattice point?
- 一定是一個原子Always one atom
- 化學鍵的交點Where bonds cross
- 電子的位置An electron position
- 一個等效位置(地址),不一定是原子An equivalent position (an address), not necessarily an atom
圖上的連線只是輔助線,並非化學鍵;晶格點是「地址」。The lines are guides, not bonds; lattice points are addresses.
地下迷宮。一份來自大阪的繞射報告完全沒有附圖,只有一串代碼:(001)、(110)、[111]、(012)。The underground maze. A diffraction report arrives with no figures, only codes: (001), (110), [111], (012).
……(hkl)。[uvw]。(米勒只用代碼說話:圓括號是平面,方括號是方向。)...(hkl). [uvw]. (Miller speaks only in code: round brackets are planes, square brackets are directions.)
五道結界擋住去路。走到結界前「調查」,解開代碼才能通過。方向不取倒數;平面先求截距、再取倒數、再化整數。Five barriers block the way. Use each one and crack its code. Directions take no reciprocals; planes go intercepts, reciprocals, integers.
(hkl) 不是標籤,是一道配方:截距、倒數、化簡、括號。(hkl) is not a label; it is a recipe: intercept, reciprocal, reduce, enclose.
指標越大,d 越小;而 d 決定繞射角。名字和間距——這就是我們的詞彙。Higher indices, smaller d; and d sets the diffraction angle. Names and spacings: that is our vocabulary.
依序調查五道結界,解出米勒指標或方向指標;清除沿路的代碼兵。Use the five barriers in order and solve the Miller or direction indices; clear the code-bots on the way.
平面:截距 → 倒數 → 最小整數 → ( )。方向:座標 → 最小整數 → [ ](不取倒數)。平面過原點要先移動原點。六方:h + k = −i。Plane: intercepts → reciprocals → integers → ( ). Direction: coordinates → integers → [ ], no reciprocals. Move the origin if the plane passes through it. Hexagonal: h + k = −i.
紅色平面與 x 軸交於 1,與 y、z 軸平行(截距 ∞)。米勒指標為?The red plane cuts x at 1 and is parallel to y and z (intercept ∞). Its Miller indices?
紅色平面的截距為 x = ½、y = 1、z = ∞。米勒指標為?Intercepts x = ½, y = 1, z = ∞. Miller indices?
紅色箭頭由原點指向 (½, 1, 0)。這個方向的指標為?(小心陷阱:選錯會走進死路!)The arrow runs from the origin to (½, 1, 0). Its direction indices? (Careful: a trap leads to a dead end!)
這個平面通過原本的原點,所以把原點移到 O′。相對 O′ 的截距為 ∞、−b、c/2。米勒指標為?The plane passes through the original origin, so the origin moves to O′. Relative to O′ the intercepts are ∞, −b, c/2. Miller indices?
六方晶系某平面已知 h = 1、k = 1、l = 0。完整的米勒–布拉維指標 (hkil) 為?A hexagonal plane has h = 1, k = 1, l = 0. The full Miller–Bravais indices (hkil)?
要先解開前面的結界!Solve the barriers in order!
結界解除!Barrier open!
你取了倒數——那是平面的程序!走進死路,被送回入口。You took reciprocals: that's the plane procedure! Dead end, back to the entrance.
代碼錯誤,結界把你彈開!Wrong code: the barrier throws you back!
真相永遠只有一個——(hkl) 不是標籤,而是一道配方:截距、倒數、化簡、括號。There is only one truth: (hkl) is not a label, it is a recipe: intercept, reciprocal, reduce, enclose.
標定方向指標 [uvw] 時,下列哪一個步驟是「錯誤」的?Which step is WRONG when indexing a direction [uvw]?
- 把座標取倒數Take reciprocals of the coordinates
- 把方向平移到通過原點Translate the line to pass through the origin
- 化為最小整數Reduce to smallest integers
- 用方括號括起Enclose in square brackets
取倒數屬於平面的程序。混用兩套步驟是最常見的錯誤。Reciprocals belong to planes; mixing the two is the most common mistake.
平面通過原點時該怎麼辦?What do you do when a plane passes through the origin?
- 把截距寫成 0Write the intercept as 0
- 把截距寫成 ∞Write it as ∞
- 平移原點後再求截距Move the origin, then find intercepts
- 這種平面無法標定It cannot be indexed
絕不可硬把截距寫成 0;移動原點就能得到漂亮的 (01̅2)。Never force a zero intercept; move the origin.
(222) 與 (111) 的關係是?How does (222) relate to (111)?
- 互相垂直Perpendicular
- 互相平行,面間距為 (111) 的一半Parallel, with half the spacing
- 完全相同的一組面,間距也相同Identical, same spacing
- 平行,但間距加倍Parallel, double spacing
(nh nk nl) 與 (hkl) 平行,間距縮為 1/n。(nh nk nl) is parallel to (hkl) with spacing 1/n.
六方晶系 (hkil) 的第三個指標 i 滿足?In hexagonal (hkil), the third index i satisfies?
- i = h + k
- i = h − k
- i = l
- i = −(h + k)
冗餘條件 h + k = −i;第四個指標帶來的是「公平」:等效晶面終於看起來也等效。Redundancy condition h + k = −i.
[hkl] 方向一定垂直於 (hkl) 平面嗎?Is [hkl] always perpendicular to (hkl)?
- 只有立方晶系才成立Only in the cubic system
- 永遠成立Always
- 永遠不成立Never
- 只有六方晶系成立Only hexagonal
此垂直關係僅在立方晶系成立,並非通則。Valid only for cubic.
呵呵,新來的偵探?我是怪盜鏡影。我能做一件「把一切都動過、卻什麼都沒改變」的事——那就是對稱。Hehe, a new detective? I'm the phantom thief Mirage. I can do something that changes everything and yet changes nothing. That is symmetry.
我用鏡面、旋轉、反轉製造分身,只有一個是本體。看出分身之間的對稱操作,到中央的鏡台「調查」,我就藏不住了。I make clones with mirrors, rotations and inversion; only one is real. Work out the symmetry operation, Use the central mirror stand, and I can't hide.
給你個提示:週期性晶體只允許 1、2、3、4、6 重旋轉。至於五重嘛……你自己判斷。A tip: periodic crystals allow only 1-, 2-, 3-, 4- and 6-fold rotation. As for five-fold... you decide.
被你看穿了。霰石與方解石是同一種化合物,只有對稱性能區分它們——而 XRD 讀的正是對稱性。You saw through me. Aragonite and calcite are the same compound; only symmetry tells them apart, and XRD reads symmetry.
結構、座標、對稱——嫌犯的完整檔案已經建立。現在,你需要一位小到能走進晶面之間的目擊者。Structure, coordinates, symmetry: the suspect's profile is complete. Now you need a witness small enough to walk between the planes.
觀察分身陣形 → 到中央鏡台判斷對稱操作 → 擊破被圈起的本體。共 7 波。Watch the formation, judge the symmetry at the mirror stand, then hit the circled original. 7 waves.
打分身只會浪費彈藥。倒立的分身代表「反轉」。五重?想想能不能鋪滿空間。Shooting clones wastes shots. Upside-down clones mean inversion. Five-fold? Can it tile space?
支線:兩座試樣台分別放著霰石(2 重)與方解石(3 重)。調查任一座,指出哪一個是方解石。Side quest: two stands hold aragonite (2-fold) and calcite (3-fold). Use either and say which is calcite.
這一波分身是由哪一種對稱操作產生的?Which symmetry operation generated this wave of clones?
鏡面 m(左右鏡射)Mirror plane m
二重旋轉(每 180° 重合)2-fold rotation (180°)
三重旋轉(每 120°)3-fold rotation (120°)
四重旋轉(每 90°)4-fold rotation (90°)
五重旋轉——週期晶體不可能,全是假的!5-fold: impossible in a crystal, all fakes!
六重旋轉(每 60°)6-fold rotation (60°)
反轉中心 i:(x, y, z) → (−x, −y, −z)Inversion centre i: (x, y, z) → (−x, −y, −z)
左邊試樣轉 120° 外觀不變,右邊試樣轉 180° 外觀不變。哪一個是方解石 CaCO₃?The left specimen repeats after 120°, the right after 180°. Which one is calcite CaCO₃?
左邊(3 重軸)Left (3-fold)
右邊(2 重軸)Right (2-fold)
第 {n} 波分身出現!Clone wave {n}!
等分身出現再判斷!Wait for the clones!
正確!五邊形無法無縫鋪滿平面,與平移對稱不相容——五重分身全是幻影。Right! Pentagons can't tile the plane; five-fold is incompatible with translation. All illusions.
本體現形了!擊破被圈起來的那一個。The original is revealed! Hit the circled one.
看錯了,分身群起攻擊!Wrong: the clones attack!
正確:方解石 3 重、霰石 2 重——化學相同、對稱不同的同質異形體。Correct: calcite 3-fold, aragonite 2-fold; same chemistry, different symmetry: polymorphs.
再看一次:轉幾度會重合?Look again: what angle brings it back?
真相永遠只有一個——對稱性就是晶體能毫髮無傷承受的操作,而週期性決定了哪些操作被允許。There is only one truth: symmetry is what a crystal can survive unchanged, and periodicity decides which operations are allowed.
週期性晶體中允許哪些旋轉軸?Which rotation axes are allowed in a periodic crystal?
- 1, 2, 3, 4, 5, 6
- 1, 2, 3, 4, 6
- 2, 4, 6, 8
- 任何 n 重皆可any n
只有 1、2、3、4、6 重能與平移對稱相容。Only 1, 2, 3, 4, 6 are compatible with translation.
為什麼晶體沒有五重旋轉軸?Why is there no five-fold axis in crystals?
- 原子數不夠Not enough atoms
- 化學鍵角限制Bond angles forbid it
- 太罕見而尚未發現Too rare to find
- 正五邊形無法無縫鋪滿空間,與平移對稱不相容Pentagons cannot tile space; incompatible with translation
鋪不滿留下的縫隙就是自白書。準晶有序但非週期,才會出現五重繞射。The gap is the confession. Quasicrystals are ordered but not periodic.
霰石(2 重)與方解石(3 重)的關係是?How are aragonite (2-fold) and calcite (3-fold) related?
- 化學組成相同、對稱不同的同質異形體Polymorphs: same chemistry, different symmetry
- 化學組成不同的兩種礦物Different chemistry
- 完全相同的晶體Identical crystals
- 一個是非晶One is amorphous
只有對稱性能區分它們,而 XRD 讀的正是對稱性。Only symmetry separates them, and XRD reads symmetry.
反轉中心 i 屬於哪一類操作?What kind of operation is an inversion centre?
- 真操作,保留掌性Proper; keeps handedness
- 平移操作A translation
- 非真操作,會反轉掌性Improper; reverses handedness
- 就是 90° 旋轉A 90° rotation
反轉把 (x,y,z) 映到 (−x,−y,−z),產生對映異構對。(x,y,z) → (−x,−y,−z) makes an enantiomorphous pair.
為什麼立方晶系的獨立繞射峰最少?Why do cubic patterns have the fewest independent peaks?
- 晶格常數最小Smallest lattice constant
- 對稱性高,許多平面變成等效High symmetry makes many planes equivalent
- 吸收最強Strongest absorption
- 原子最少Fewest atoms
對稱越多、線索越少;最對稱的晶體也最容易標定。More symmetry, fewer clues, and easier indexing.
我是夜鷹 λ。晶面之間只隔 0.2–0.5 nm,沒有透鏡能解析。我們需要一位波長和縫隙一樣大的目擊者:X 光。I'm Nighthawk λ. Planes are only 0.2–0.5 nm apart; no lens can resolve them. We need a witness whose wavelength matches the gap: X-rays.
從現在起,自動瞄準關閉。晶體只在「額外路徑恰為整數個波長」的角度開口:λ = 2d sinθ。From now on, no auto-aim. A crystal speaks only where the extra path is a whole number of wavelengths: λ = 2d sinθ.
Cu Kα 的 λ = 0.15418 nm,石英靶 d = 0.334 nm。算出 2θ 調好刻度:E/Q 微調、C/Z ±1°、滑鼠滾輪,或按 Enter(手機點刻度)直接輸入。Cu Kα: λ = 0.15418 nm; the quartz target has d = 0.334 nm. Compute 2θ and set the dial: E/Q fine, C/Z ±1°, the wheel, or Enter (tap the dial on a phone) to type it.
命中!2θ ≈ 26.7°。現在換彈種:Mo Kα(0.07107 nm)與 Cr Kα(0.22910 nm)。Hit! 2θ ≈ 26.7°. Now switch: Mo Kα (0.07107 nm) and Cr Kα (0.22910 nm).
同一個 d,λ 一變,角度就跟著動。這些靶只接受指定的彈種。Same d, new λ, new angle. These targets accept only the named tube.
最後一個靶:d = 0.070 nm。先試 Cu 和 Cr——注意畫面。Last target: d = 0.070 nm. Try Cu and Cr first, and watch the screen.
sinθ 不可能大於 1,所以必須 λ < 2d。這裡 2d = 0.14 nm,只有短波長的 Mo 打得到。sinθ can never exceed 1, so λ < 2d. Here 2d = 0.14 nm: only short-wavelength Mo can reach it.
漂亮。粉末裡有數百萬顆隨機取向的晶粒:對每一組 (hkl),總有些晶粒恰好在布拉格角,繞射線排成半頂角 2θ 的圓錐。Nicely done. A powder holds millions of randomly oriented crystallites; for every (hkl) some sit exactly at the Bragg angle, and their beams form a cone of half-angle 2θ.
隨機取向不是麻煩,而是保證「每一個晶面都有機會出庭作證」。石英兵來了,它們身上不只一組晶面——挑一組算。Random orientation is not a nuisance; it guarantees every plane gets to testify. Quartz soldiers incoming, each with several planes: pick one and compute.
峰位告訴我們「在哪裡」。但兩種化合物可能共用同一個 d 值——光靠峰位無法定罪,我們還需要峰高。Peak positions tell us WHERE. But two compounds can share a d-spacing; position alone convicts no one. We need the heights.
用 Cu 彈擊中三個石英靶(d = 0.334 nm)。Hit the three quartz targets (d = 0.334 nm) with Cu.
2θ = 2·arcsin(λ / 2d);手機可點 2θ 刻度直接輸入數字。2θ = 2 arcsin(λ / 2d); on a phone, tap the dial to type a number.
換彈種:用 Mo 擊中 Mo 靶、用 Cr 擊中 Cr 靶(d 仍是 0.334 nm)。Switch tubes: Mo targets with Mo, Cr targets with Cr (d still 0.334 nm).
找出能命中 d = 0.070 nm 小靶的彈種。Find the tube that can hit the d = 0.070 nm target.
擊破石英兵(任選彈種與晶面)。Defeat the quartz soldiers (any tube, any plane).
真相永遠只有一個——晶體只在「額外路徑恰為整數個波長」的角度上開口說話。There is only one truth: the crystal answers at exactly those angles where the extra path equals a whole number of wavelengths.
Cu Kα(λ = 0.15418 nm)照射 d = 0.334 nm 的石英 (101),2θ 約為?Cu Kα (λ = 0.15418 nm) on quartz (101), d = 0.334 nm: 2θ is about?
- 13.3°
- 26.7°
- 31.1°
- 53.4°
sinθ = 0.15418 / 0.668 = 0.231,θ = 13.3°,2θ = 26.7°。31.1° 是 Co 靶的答案。sinθ = 0.231, θ = 13.3°, 2θ = 26.7° (31.1° would be Co).
為什麼 d = 0.070 nm 的晶面只有 Mo 打得到?Why can only Mo reach a plane with d = 0.070 nm?
- 必須 λ < 2d = 0.14 nm,只有 Mo(0.0711 nm)符合Need λ < 2d = 0.14 nm; only Mo (0.0711 nm) qualifies
- Mo 的強度最高Mo is the most intense
- Cu 和 Cr 會產生螢光Cu and Cr fluoresce
- Mo 的濾片比較好Mo has a better filter
sinθ ≤ 1 ⇒ λ < 2d。sinθ ≤ 1 ⇒ λ < 2d.
同一個 d,改用波長較長的 Cr 靶,2θ 會?Same d, switch to longer-wavelength Cr: 2θ will?
- 變小Decrease
- 不變Stay the same
- 變大Increase
- 一定消失Always vanish
sinθ = λ/2d,λ 變大則 θ 變大(d = 0.334 nm 時 Cr ≈ 40.1°)。sinθ = λ/2d grows with λ (Cr ≈ 40.1° here).
粉末法為什麼得到的是一維圖譜?Why is a powder pattern one-dimensional?
- 因為偵測器只有一個像素The detector has one pixel
- 因為晶體只有一個 d 值Crystals have one d
- 因為 X 光只能走直線X-rays travel straight
- 隨機取向的晶粒形成繞射錐,偵測器沿圓弧掃過記錄強度對 2θRandom grains form diffraction cones; the detector scans an arc recording I vs 2θ
隨機晶粒把三維問題變成沿繞射錐的一維掃描。Random grains turn 3-D into a 1-D scan of cones.
為什麼用 X 光而不是可見光做繞射?Why X-rays rather than visible light?
- X 光波長(約 0.05–0.25 nm)與面間距同一數量級X-ray wavelengths (~0.05–0.25 nm) match interplanar spacings
- X 光可以用透鏡聚焦X-rays focus with lenses
- X 光帶電X-rays are charged
- 可見光太危險Visible light is dangerous
這個「巧合」正是 XRD 存在的全部理由;X 光不帶電、無質量、也難以聚焦。That coincidence is why XRD exists; X-rays are uncharged and hard to focus.
我是霜月,強度分析師。兩張圖譜峰位完全相同,峰高卻截然不同;其中一張裡,定律說「必須存在」的峰卻消失了。I'm Shizuki, intensity analyst. Two patterns with identical positions but different heights; and in one, a peak the law demands has vanished.
鐵衛與銅衛外觀相同、護盾也相同。逐一試射 (100)、(110)、(111)、(200),記錄哪些有反應、哪些「免疫」。The Iron and Copper Guardians look alike and share a shield. Test-fire (100), (110), (111), (200) and note which respond and which are immune.
消失的峰不是證據不見了,它本身就是證據。每位至少試三個晶面後,到終端機判斷晶格型式。A missing peak is not missing evidence; it IS evidence. After testing at least three planes on each, judge the lattice types at the terminal.
判斷正確,弱點開放!再提醒一件事:原子散射因子 f 隨 sinθ/λ 遞減,高角度命中的傷害自然比較低。Correct: weak points open! One more thing: the scattering factor f falls with sinθ/λ, so high-angle hits deal less.
而且戰鬥越久溫度越高:原子振動加劇、圖譜模糊、峰值再降(溫度因子 e−2M)。速戰速決。And the longer we fight, the hotter it gets: vibration grows, the pattern smears, peaks drop (temperature factor e−2M). Be quick.
峰位鑑定晶格,峰強鑑定晶格裡裝了什麼,而消失的峰,是對稱性的自白。Positions identify the lattice, intensities what sits inside, and a missing peak is a confession about symmetry.
第一章結案。下一章,儀器本身也將成為嫌犯。Chapter one is closed. Next, the instrument itself becomes a suspect.
對兩位守衛試射 (100)、(110)、(111)、(200),各至少三個晶面;再到終端機判斷晶格型式。Test-fire (100), (110), (111), (200) on both guardians, at least three each, then judge the lattices at the terminal.
d = a / √(h² + k² + l²)。目標卡會記錄「有峰/無峰」。d = a / √(h² + k² + l²). The target card records peak / no peak.
擊敗鐵衛與銅衛(高角度傷害較低)。Defeat both guardians (high-angle hits deal less).
鐵衛(a = 0.2866 nm)的試射結果如圖。它的晶格型式是?The Iron Guardian (a = 0.2866 nm) test results are shown. Its lattice type?
銅衛(a = 0.3615 nm)的試射結果如圖。它的晶格型式是?The Copper Guardian (a = 0.3615 nm) test results are shown. Its lattice type?
證詞不足:每位守衛至少要試射三個不同晶面。Not enough testimony: test at least three planes on each guardian.
判斷錯誤,護盾反震!Wrong: the shield recoils!
溫度上升(第 {n} 級):圖譜模糊,傷害下降。Temperature up (level {n}): pattern smears, damage drops.
溫度因子作用中:第 {n} 級Temperature factor: level {n}
真相永遠只有一個——峰位鑑定晶格,峰強鑑定晶格裡裝了什麼,而消失的峰,是對稱性的自白。There is only one truth: peak positions identify the lattice, peak intensities identify what sits inside it, and a missing peak is a confession about symmetry.
體心 I 晶格會出現哪些反射?Which reflections appear for a body-centred (I) lattice?
- 全部 (hkl)All (hkl)
- h、k、l 全奇或全偶h, k, l all odd or all even
- h + k + l 為偶數h + k + l even
- h + k 為偶數h + k even
α-Fe:(110) 44.7°、(200) 65.0°、(211) 82.3°;(100)、(111) 消光。α-Fe shows (110), (200), (211); (100) and (111) are absent.
面心 F 晶格會出現哪些反射?Which reflections appear for a face-centred (F) lattice?
- h + k + l 為偶數h + k + l even
- h、k、l 全奇或全偶h, k, l all odd or all even
- 全部 (hkl)All (hkl)
- 只有 (h00)Only (h00)
Cu:(111) 43.4°、(200) 50.5°、(220) 74.1°;(100)、(110) 消光。Cu shows (111), (200), (220); (100) and (110) are absent.
α-Fe 為何沒有 (100) 峰?Why does α-Fe show no (100) peak?
- 體心原子在正中間插入一層反相的平面,F = 0 完全抵消The body-centre atom inserts an out-of-phase plane; F = 0 cancels it
- 波長太短Wavelength too short
- 吸收太強Absorption too strong
- 晶粒太細Grains too fine
相差半個波長且散射能力相等 → 完全抵消。Half a wavelength out of step with equal scattering → total cancellation.
原子散射因子 f 隨繞射角如何變化?How does the atomic scattering factor f change with angle?
- 遞增Increases
- 不變Constant
- 先增後減Rises then falls
- 遞減(2θ = 0 時 f = Z),所以高角度峰本質較弱Decreases (f = Z at 2θ = 0), so high-angle peaks are weak
原子內不同電子的路徑差隨角度增大而失去同步。Electrons within the atom fall out of step as the angle opens.
立方晶系 {111} 的多重性因子 p 是?The multiplicity p of cubic {111} is?
- 6
- 12
- 8
- 24
±h ±k ±l 的排列:{111} 為 8,{100} 為 6。{111} gives 8, {100} gives 6.
若插入面的原子與原平面的原子不同(f₁ ≠ f₂),原本消光的峰會?If the inserted atoms differ (f₁ ≠ f₂), the extinct peak will?
- 依然完全消失Still vanish
- 以微弱的峰回來(如有序合金的超晶格線)Return weakly (superlattice lines in ordered alloys)
- 變得最強Become strongest
- 移到別的角度Move to another angle
抵消需要「相等」;打破相等,消失的峰就以低語回來。Cancellation needs equality; break it and the peak whispers back.
第二章!歡迎來到我的工坊——偵訊室。一盞你能控制的燈、一個能旋轉的對象,以及一個固定距離的證人席。Chapter two! Welcome to my workshop, the interrogation room: a lamp you control, a subject you can turn, a witness stand at a fixed distance.
Bragg–Brentano θ–2θ:試樣以偵測器一半的角速度旋轉,表面始終與聚焦圓相切,繞射線收斂在接收狹縫上。Bragg–Brentano θ–2θ: the sample turns at half the detector's speed, staying tangent to the focusing circle so the beam converges on the receiving slit.
每一項設定都是一次取捨。我們一項一項解鎖,先從光管開始:管電壓決定電子多快,管電流決定電子多少。Every setting is a trade-off. We unlock them one at a time, starting with the tube: kV sets how fast the electrons are, mA how many.
過熱停機!輸入功率有 99% 以上變成熱——功率高、傷害高,但一定要顧散熱(實機靠冷卻水)。Overheated! Over 99% of the input power becomes heat. More power, more damage, but mind the cooling (real tubes need cooling water).
下一項:接收狹縫,也就是你的瞄準鏡。切到「窄」,觀察容許範圍與傷害怎麼變。Next: the receiving slit, your scope. Switch it to Narrow and watch the tolerance and damage change.
看到了嗎?窄狹縫容許範圍小、能分辨相近的峰,但傷害下降。解析度與強度在同一根槓桿的兩端。See? A narrow slit tightens the tolerance and separates close peaks, but costs damage. Resolution and intensity sit on opposite ends of one lever.
下一項:索勒狹縫——一疊平行薄片,限制軸向發散,峰形更對稱。裝上它再射一發。Next: Soller slits, a stack of parallel foils that limit axial divergence for more symmetric peaks. Fit them and fire again.
好。下一項:β 濾片。先「不裝」射幾發,看下方圖譜多出的藍色虛線——那是 Kβ 幽靈峰,出現在較小的 2θ。Good. Next: the β filter. Fire a few shots WITHOUT it first and look at the blue dashed lines on the pattern: Kβ ghosts at lower 2θ.
然後裝上濾片(Cu 用 Ni):它的吸收邊剛好落在 Kα 與 Kβ 之間,專門吞掉 Kβ。Then fit the filter (Ni for Cu): its absorption edge sits between Kα and Kβ and swallows Kβ.
一片濾片、一道吸收邊,除掉一條不要的譜線。最後一項:單光儀——一片固定在 Kα 布拉格角上的晶體。One foil, one edge, one unwanted line gone. Last: the monochromator, a crystal set at the Kα Bragg angle.
單光儀同時除掉 Kβ 與試樣螢光,這是濾片做不到的;代價是強度再一次大幅下降。The monochromator removes both Kβ and specimen fluorescence, which a foil cannot; the price is another large drop in intensity.
警報!高鐵爐石兵入侵!注意:Fe 在 Cu Kα 照射下會產生螢光——背景升高、峰被淹沒。Alarm! Iron-rich slag soldiers! Careful: Fe fluoresces under Cu Kα; the background rises and the peaks drown.
兩種硬體解法:改用 Co 或 Fe 靶(剛剛解鎖了),或裝上單光儀。Two hardware fixes: switch to a Co or Fe tube (just unlocked), or fit the monochromator.
最後的挑戰:一對雙生石英兵,峰位只差 0.05°。單一設定分不開它們。Final test: twin quartz soldiers whose peaks differ by only 0.05°. No single setting separates them.
窄狹縫 + 索勒 + 長時間蓄力(按住射擊:計數 N 越大,精度 ∝ √N)。三者齊備才能分開擊破。Narrow slit + Soller + a long charge (hold fire: precision ∝ √N). All three together separate them.
繞射儀從不單純記錄晶體,它記錄的是「透過你所選光路看到的晶體」。A diffractometer never just records the crystal; it records the crystal as seen through the optics you chose.
但人手蓋的房間可能蓋歪一點點——而一間歪了一點點的房間,會產出一份極具說服力的假供詞。But a room built by hand can be slightly crooked, and a slightly crooked room produces a perfectly convincing false confession.
把功率調到 3,對訓練靶連續射擊,直到過熱停機。Set power to 3 and fire at the training target until it overheats.
把接收狹縫切到「窄」,命中訓練靶。Set the receiving slit to Narrow and hit the target.
裝上索勒狹縫,命中訓練靶。Fit the Soller slits and hit.
先不裝濾片射擊(看 Kβ 假峰),再裝上 β 濾片命中。Fire without the filter (see the Kβ ghosts), then fit the β filter and hit.
裝上單光儀,命中訓練靶。Fit the monochromator and hit.
擊敗高鐵爐石兵(避開螢光霧:換 Co/Fe 靶,或用單光儀)。Defeat the slag soldiers (avoid the fluorescence fog: Co/Fe tube or the monochromator).
磁鐵礦 (311) d = 0.2532 nm。換靶材後記得重算 2θ。Magnetite (311) d = 0.2532 nm. Recompute 2θ after changing tube.
分開擊破雙生石英兵:窄狹縫 + 索勒 + 蓄力到 N = 4。Separate the twins: narrow slit + Soller + charge to N = 4.
按住「射擊」蓄力,放開才發射;蓄力條全滿 = N 4。兩者 d = 0.3343 與 0.3337 nm。Hold FIRE to charge and release to shoot; a full bar = N 4. d = 0.3343 and 0.3337 nm.
命中:容許範圍 {tol},傷害 {dmg}Hit: tolerance {tol}, damage {dmg}
真相永遠只有一個——繞射儀從不單純記錄晶體,它記錄的是「透過你所選光路看到的晶體」。There is only one truth: the diffractometer never simply records the crystal; it records the crystal as seen through the optics you selected.
要分辨相距 0.05° 2θ 的兩根峰,應怎麼調整?代價是什麼?To resolve two peaks 0.05° 2θ apart, what do you change, and at what cost?
- 提高 kV 就好,沒有代價Raise kV; no cost
- 換寬狹縫增加強度Use a wider slit for intensity
- 拿掉濾片Remove the filter
- 窄接收狹縫、加索勒、延長計數或放慢掃描——代價是強度與時間Narrow receiving slit, Soller slits, longer counting / slower scan; paid in counts and time
解析度的每一分提升,都要用計數或時間來付帳。Every gain in resolution is paid for in counts or time.
實驗室只有 Cu 靶,試樣是高含鐵量的爐石。圖譜會怎樣?如何解決?Only a Cu tube, and an iron-rich slag. What will the pattern look like, and the fixes?
- Fe 螢光使背景升高、峰被淹沒;改用 Co/Fe 靶或加單光儀Fe fluorescence raises the background; use a Co/Fe tube or a monochromator
- 峰會全部消失;改用 Mo 靶All peaks vanish; use Mo
- 完全正常Perfectly normal
- 只會出現 Kβ;加 Ni 濾片即可Only Kβ appears; a Ni filter fixes it
Cu Kα 能量高於 Fe 的 K 吸收邊而激發螢光;Ni 濾片去不掉試樣螢光。Cu Kα is above the Fe K edge; a Ni foil cannot remove specimen fluorescence.
β 濾片與單光儀最大的差別是?The key difference between a β filter and a monochromator?
- 兩者完全相同They are the same
- 單光儀還能去除試樣螢光,但強度損失更大The monochromator also removes specimen fluorescence, at a bigger intensity cost
- 濾片能去除螢光The filter removes fluorescence
- 單光儀會增加強度The monochromator adds intensity
單光儀是固定在 Kα 布拉格角上的晶體。A monochromator is a crystal set at the Kα Bragg angle.
Bragg–Brentano 幾何中,試樣的角速度是偵測器的?In Bragg–Brentano geometry the sample turns at what fraction of the detector speed?
- 相同Same
- 兩倍Double
- 一半(θ–2θ 連動)Half (θ–2θ)
- 試樣不動The sample is fixed
如此試樣表面在任何角度都與聚焦圓相切。So the surface stays tangent to the focusing circle.
X 光管的輸入功率大約有多少變成熱?Roughly how much of an X-ray tube's power becomes heat?
- 超過 99%Over 99%
- 50%
- 10%
- 1%
九成九是熱,必須以冷卻水降溫;轉靶可把熱分散到更大面積。Almost all heat; hence cooling water, or a rotating anode.
呵呵,你的槍我動過了。每一發彈道都偏 +0.06°,而且我把接收狹縫鎖在最精密的校準檔(±0.04°)。Hehe, I've touched your gun. Every shot is now off by +0.06°, and I've locked your slit in the precision calibration setting (±0.04°).
證人不需要存心欺騙,也足以把你帶偏。去石英標準靶檢查峰位,找出是哪一種誤差吧。A witness need not lie to mislead you. Check peak positions on the quartz standard and work out which error it is.
零點偏移校正好了?可是再量一次——峰還是偏,而且偏移量隨角度改變,低角度最大。Zero corrected? Measure again: the peaks still shift, and the shift changes with angle, largest at low angle.
這次不是儀器零點,而是你的「樣品高度」。試樣表面只要偏離轉軸零點幾毫米,聚焦條件就失效。This time it is not the zero but your sample height. A surface a fraction of a millimetre off the axis loses the focusing condition.
校正完成,狹縫解鎖。最後一關:氫氧化鈣盾兵——六方板狀晶體,整隊以 (001) 面朝外排列。Calibrated; slit unlocked. Last: the portlandite shield squad, hexagonal plates all lined up with (001) facing you.
(001) 護盾強到打不穿,其他晶面卻幾乎打不痛——擇優取向會改寫相對強度。用「旋轉試樣」打散排列,或到工作台改用側向裝填。The (001) shield won't break and the other planes barely hurt: preferred orientation rewrites relative intensity. Spin the sample, or switch to side-loading at the bench.
一張繞射圖譜同時量到了晶體、儀器,以及你填裝粉末的方式。下一案見——我會一直在旁邊看著。A pattern measures the crystal AND the instrument AND how you packed the powder. See you next case; I'll be watching.
在石英標準靶上量測至少兩個相隔 15° 以上的峰,再到校正台診斷誤差。Measure at least two quartz-standard peaks 15°+ apart, then diagnose at the calibration desk.
射向標準靶,只要在 ±0.5° 內就會記錄峰位。關鍵問題:位移是否隨角度而變?Shots within ±0.5° record a peak. Ask: does the shift depend on angle?
再量一次(至少兩個相隔 15° 以上的峰),診斷剩下的誤差並修正。Measure again (two peaks 15°+ apart), diagnose the remaining error and fix it.
加量一個高角度峰(d = 0.1082 nm,Cu ≈ 90.8°)會更清楚。Add a high-angle peak (d = 0.1082 nm, Cu ≈ 90.8°) to see it clearly.
擊敗氫氧化鈣盾兵:用「旋轉試樣」(R),或到工作台改用側向裝填。Defeat the portlandite squad: spin the sample (R), or side-load at the bench.
石英標準靶的量測結果如表。最可能的誤差來源是?The quartz-standard results are tabulated. The most likely error?
零點已校正,再量一次的結果如表。剩下的誤差來源是?With the zero corrected, the new results are tabulated. The remaining error?
試樣位移應該如何修正?How do you fix specimen displacement?
2θ 零點偏移:所有峰同量、同向位移,與角度無關2θ zero offset: all peaks shift equally, angle-independent
試樣位移:偏移隨角度而變(∝ cosθ),低角度最大Specimen displacement: angle-dependent (∝ cosθ), largest at low angle
擇優取向:改變相對強度Preferred orientation: changes relative intensity
Kβ 假峰Kβ ghost peaks
重新調整試樣高度,讓表面落在測角器轉軸上Reset the sample height so the surface sits on the goniometer axis
再改一次零點Change the zero again
換更寬的狹縫Use a wider slit
至少要兩個峰,而且角度要相差 15° 以上,才看得出位移是否隨角度改變。You need two peaks at least 15° apart to tell whether the shift depends on angle.
診斷錯誤。看看表中的 Δ2θ 是否隨角度改變。Wrong diagnosis. Does Δ2θ change with angle?
(001) 異常強、其他晶面異常弱——這就是擇優取向!片狀晶粒傾向以片面平行於表面躺平。(001) far too strong, the rest far too weak: preferred orientation! Plates lie flat on the surface.
改用側向裝填:晶面不再被壓貼在表面上,擇優取向大幅減少。Side-loading: faces are no longer pressed onto the surface; orientation drops sharply.
真相永遠只有一個——一張繞射圖譜同時量到了晶體、儀器,以及你填裝粉末的方式。There is only one truth: a diffraction pattern is a measurement of the crystal AND of the instrument AND of how you packed the powder.
同一批熟料量兩次:A 組所有峰位移 +0.06°;B 組只有低角度峰位移。請診斷。Two runs: A has every peak shifted +0.06°; B shifts only at low angle. Diagnose.
- A 試樣位移;B 零點偏移A displacement; B zero offset
- A 零點偏移;B 試樣位移A zero offset; B displacement
- 兩者都是擇優取向Both preferred orientation
- 兩者都是 KβBoth Kβ
先問:位移是否隨角度而變?與角度無關 = 儀器零點;與角度有關 = 試樣位移。可用石英標準片驗證。Angle-independent = instrument zero; angle-dependent = specimen displacement. Verify with a quartz standard.
硬固漿體中 CH (001) 遠高於卡片、其他譜線卻偏低。原因與改善?CH (001) is far stronger than the card and the others too weak. Cause and fixes?
- 零點偏移;重新歸零Zero offset; re-zero
- Kβ 干擾;加濾片Kβ; add a filter
- CH 含量太多;稀釋Too much CH; dilute
- 擇優取向(六方板狀);改用側/背向裝填、旋轉試樣或細化粒徑Preferred orientation (hexagonal plates); side/back-load, spin, or grind finer
相對強度的前提是隨機取向;片狀晶粒會躺平。Relative intensity assumes random orientation; plates lie flat.
用石英標準片要檢查哪三件事?Which three things do you check with a quartz standard?
- 峰位、峰強、角解析度(五重峰)Position, intensity, angular resolution (quintuplet)
- 顏色、重量、硬度Colour, weight, hardness
- 只有峰位Position only
- 冷卻水溫度Cooling-water temperature
分不開石英五重峰的儀器,也分不開你的相。If it can't split the quintuplet, it can't split your phases.
可變發散狹縫的代價是?What is the price of a variable divergence slit?
- 峰位全部偏移All peaks shift
- 產生 KβIt adds Kβ
- 相對強度被改寫,與 PDF 卡片比對前必須換算Relative intensities change; convert before comparing with PDF cards
- 無法量低角度No low-angle data
它讓試樣受照面積維持不變,避免低角度溢射到樣品座。It keeps the irradiated area constant and stops low-angle spill.
我是檔案官。桌上這隻「十九峰合成體」沒有化學式、沒有標籤、沒有來歷。I'm the Archivist. This Nineteen-Peak Chimera has no formula, no label, no history.
流程:先到掃描台執行 θ–2θ 掃描,記錄峰位與強度;再到資料庫終端機,用最強的兩根峰搜尋 PDF 卡片。The method: run a θ–2θ scan at the scan desk to record positions and intensities; then search the PDF cards at the database terminal with the two strongest peaks.
確認一個相,就把它的峰全部扣除,合成體失去那個部位。黃金守則:每一根峰都必須被指派。Confirm a phase and subtract all its peaks; the chimera loses that part. The golden rule: assign every peak.
等等!你接受了含鋇的相?這份試樣只用了水泥、水和砂——鋇從哪裡來?Wait! You accepted a barium phase? This sample was made from cement, water and sand. Where would barium come from?
92% 的吻合分數毫無意義。化學組成與製程履歷優先於比對分數——打開「化學篩選」。合成體回滿血了……A 92% score means nothing. Chemistry and processing history outrank any score: turn on the chemistry filter. The chimera is back at full health...
相鑑定不是找到一個好的吻合,而是把圖譜上的每一根峰都交代清楚。Identification is not finding one good match; it is accounting for every peak in the pattern.
我們能指認出各個相了。下一章要問的是每位工程師真正想問的:各有多少?We can name the phases. Next chapter asks what every engineer really asks: how much of each?
掃描台掃描 → 資料庫終端機 search–match → 接受正確的相;指派所有峰(含 Kβ 假峰)。Scan → search–match at the database → accept the right phases; assign every peak, including Kβ ghosts.
沒裝濾片時掃描會混入 Kβ 假峰:可在資料庫標記,或裝濾片後重掃。注意卡片品質評等(* 最可靠)。Without a filter the scan adds Kβ ghosts: mark them in the database or rescan with the filter. Check card quality (* is best).
θ–2θ 掃描中……保護好自己!θ–2θ scan in progress... stay alive!
還沒有任何峰紀錄。先到掃描台掃描。No peaks recorded. Scan first.
掃描完成:峰位與強度已記錄。Scan complete: positions and intensities recorded.
掃描完成——但沒裝濾片,圖譜混入了 Kβ 假峰(藍色虛線)。Scan complete, but without a filter Kβ ghosts (blue dashed) slipped in.
正確:那是 Kβ 假峰,已刪除。Correct: a Kβ ghost, removed.
那是真正的峰!別把證據丟掉。That's a real peak! Don't throw evidence away.
{n} 並不在這份試樣中:合成體恢復了部分體力。{n} is not in this sample: the chimera recovers.
這個相已經指派過了。Already assigned.
指派成功,但你用了年代久遠、單一來源的低品質卡片——推理分扣分。下次先讀量測條件。Assigned, but from an old single-source card; reasoning points lost. Read the conditions first.
確認:{n}!其所有峰已扣除。Confirmed: {n}! All its peaks subtracted.
還有未指派的峰!合成體從那根峰重生了。Unassigned peaks remain! The chimera regenerates from them.
真相永遠只有一個——相鑑定不是找到一個好的吻合,而是把圖譜上的每一根峰都交代清楚。There is only one truth: identification is not finding one good match, it is accounting for every peak in the pattern.
28 天漿體已指派 CH、鈣礬石、方解石與殘餘 alite,仍有四根中等峰無法解釋。最先該做什麼?A 28-day paste is assigned to CH, ettringite, calcite and alite, but four medium peaks remain. What first?
- 先做最省事的檢查:是否為 Kβ、樣品座等假峰,再依序考慮次要相、晶型與化學萃取Cheapest first: Kβ or holder artefacts, then minor phases, polymorphs, extractions
- 直接接受分數最高的卡片Accept the top-scoring card
- 當成雜訊忽略Ignore them as noise
- 提高 kV 重新量測Raise kV and remeasure
先做最省事的檢查,不是最聰明的那個。未被指派的峰,在證明它是雜訊之前都不算雜訊。Start with the cheapest check, not the cleverest.
你自己用 OPC、水、砂拌製的試樣,軟體給出含鋇相 92% 吻合。怎麼做?Your own OPC/water/sand sample returns a barium phase at 92%. What do you do?
- 接受,因為分數很高Accept: the score is high
- 提高容許範圍再搜尋Widen the window
- 拒絕:鋇不可能來自原料,化學組成與製程履歷優先於分數Reject: barium cannot come from the materials; chemistry outranks the score
- 改用 Mo 靶重量Remeasure with Mo
先問:鋇到底可能從哪裡來?Ask where the barium could possibly come from.
Search–match 的起點是?Where does search–match start?
- 最弱的峰The weakest peaks
- 以最強兩根峰,設定容許範圍搜尋The two strongest peaks, with a tolerance window
- 任挑一根Any one peak
- 先猜化學式Guess the formula
強度本身不可靠(擇優取向),所以要用容許範圍而非精確值。Intensities are unreliable, so use a window, not an exact value.
PDF 卡片上的星號(*)代表?A star (*) on a PDF card means?
- 最強峰Strongest peak
- 有毒物質Toxic
- 年代最久Oldest
- 資料完整、可靠Well characterised, reliable
優先選用有星號、年代較新、多來源的條目;早期資料多半不可靠。Prefer starred, recent, multi-source entries.
相鑑定的黃金守則是?The golden rule of phase identification?
- 每一根峰都必須被指派、被標定Assign and index every peak
- 只看最強峰Only the strongest peak matters
- 分數 > 90% 就接受Accept anything over 90%
- 弱峰都是雜訊Weak peaks are noise
該有的就應該有。Everything present must be accounted for.
第三章。三個房間各有一團不同的煙霧(基質),裡面都藏著恰好 50 wt% 的石英兵。Chapter three. Three rooms, each with a different smoke (matrix), each hiding quartz at exactly 50 wt%.
石英從頭到尾都沒有說謊。是站在光束與石英之間的東西投下了陰影——陰影大小,完全取決於房間裡還有誰。The quartz never lied. Something between the beam and the quartz casts a shadow, and its size depends entirely on who else is in the room.
在每個房間打中石英兵、讀出峰高,再到該房前方的終端機回報「真實含量」,不是讀數。Hit the quartz in each room, read the peak height, then report the TRUE content, not the reading, at that room's terminal.
給你「內標信標」。在房間放入固定量的已知相:石英與內標處在「同一個」基質中,讀數就能換算回真實值。去各房間啟動它。Here are internal-standard beacons. Add a fixed amount of a known phase: quartz and standard share the SAME matrix, so the reading converts back to the truth. Switch them on in each room.
最後一間:石英與含鐵重相 50/50 混合。開槍之前,先預測。Last room: quartz mixed 50/50 with a heavy iron-bearing phase. Predict before you shoot.
峰高回報的,是含量除以「試樣中其他所有東西的吸收」。唯有兩相吸收相同時,強度才真正正比於含量。Peak height reports content divided by the absorption of everything else. Only identical absorbers give intensity proportional to content.
基質會扭曲答案。而粉末的製備方式,同樣能把答案扭曲得一樣嚴重。The matrix distorts the answer, and so can the way you prepare the powder.
三個房間各打中石英兵讀取峰高,到該房終端機回報真實含量;清除所有石英兵。In each room, hit quartz to read the peak height, report the true content at that terminal, and clear all quartz.
Iα ∝ wα(μ/ρ)α / Σwi(μ/ρ)i。Cu Kα 下 μ/ρ:石英 ≈ 36、BeO ≈ 9(弱)、KCl ≈ 122(強)、方矽石 = 石英。I ∝ w(μ/ρ)/Σw·μ/ρ. With Cu Kα: quartz ≈ 36, BeO ≈ 9 (weak), KCl ≈ 122 (strong), cristobalite = quartz.
含鐵重相房:啟動內標信標並擊敗石英兵。Iron-phase room: switch on the beacon and defeat the quartz.
這個房間石英的峰高讀數是 {p}%。石英的真實含量是?Here the quartz peak reads {p}%. What is the true quartz content?
石英與含鐵重相 50/50 混合。石英峰會高於、還是低於 50% 基準線?Quartz mixed 50/50 with a heavy iron phase: will the quartz peak read above or below the 50% line?
高於 50%Above 50%
低於 50%:鐵相吸收強,平均 μ/ρ 變大Below 50%: the iron phase raises the mean μ/ρ
剛好 50%Exactly 50%
讀數(峰高):{p}%Reading (peak height): {p}%
內標換算:讀數 {p}% → 真實 50%Internal standard: {p}% read → 50% true
先在這個房間打中石英兵,取得讀數。First hit a quartz soldier in this room for a reading.
內標信標啟動:此區讀數換算回真實含量,傷害恢復正常。Beacon on: readings here convert to true content; damage back to normal.
預測正確:重相吸收強,石英看起來比實際少(約 15%)。開槍驗證吧!Correct: the heavy phase absorbs strongly, so quartz looks scarcer (~15%). Shoot to verify!
再想想:鐵相讓平均 μ/ρ 變大,石英看起來更少(約 15%)。開槍驗證吧!Think again: iron raises the mean μ/ρ, so quartz reads lower (~15%). Shoot to verify!
那是讀數,不是含量。每個房間都是 50 wt%,差別只在基質吸收。That's the reading, not the content. Every room is 50 wt%; only the matrix absorption differs.
真相永遠只有一個——峰高所回報的,是含量除以試樣中其他所有東西的吸收。There is only one truth: a peak height reports concentration divided by the absorption of everything else in the specimen.
多相試樣中,某相的峰高反映的是?In a multiphase specimen, a phase's peak height reflects?
- 只有該相含量Its content only
- 含量除以整個基質的平均吸收Content divided by the mean absorption of the whole matrix
- 只有基質吸收Matrix absorption only
- 晶粒大小Grain size
Iα ∝ φα / μ̄*:換了基質、換了分母,答案就變了,而含量根本沒動。Change the matrix, change the denominator, change the answer.
50% 石英與 50% 弱吸收體(BeO,μ/ρ ≈ 9)混合,石英峰相對於純石英會?50% quartz with 50% weakly absorbing BeO (μ/ρ ≈ 9): relative to pure quartz, the quartz peak reads?
- 高於 50%(約 80%):基質讓光束更深入試樣Above 50% (~80%): the matrix lets the beam reach more quartz
- 低於 50%(約 25%)Below 50% (~25%)
- 剛好 50%Exactly 50%
- 完全看不到Invisible
I/I純 = 0.5×36 / (0.5×36 + 0.5×9) ≈ 0.8;強吸收體 KCl 則降到約 0.23。這就是 Cullity Fig. 7-4 中石英–BeO 曲線在對角線上方、石英–KCl 在下方的原因。0.5×36 / (0.5×36 + 0.5×9) ≈ 0.8; with KCl it drops to ~0.23 (quartz–BeO above the diagonal, quartz–KCl below).
內標法依賴的關鍵假設是?The key assumption behind the internal-standard method?
- 內標比待測相多More standard than analyte
- 內標一定是石英The standard is quartz
- 不需要研磨No grinding needed
- 待測相與內標處在同一個基質中,受到相同的吸收Analyte and standard sit in the same matrix and see the same absorption
兩者比值與基質吸收無關,所以能換算回真實含量。Their ratio no longer depends on the matrix.
為什麼石英–方矽石混合物落在對角線上?Why does quartz–cristobalite lie on the diagonal?
- 兩者顏色相同Same colour
- 兩者晶系相同Same crystal system
- 兩者吸收係數完全相同(同為 SiO₂)Identical absorption (both SiO₂)
- 兩者峰位相同Same peak positions
吸收相同時,強度才真正正比於含量。Only identical absorbers give a straight line.
又見面了。這次沒有人是故意動手腳的——這才危險。製樣看起來只是「照程序走」,所以沒人想到要質疑它。We meet again. Nobody tampers on purpose this time, which is what makes it dangerous. Preparation looks like procedure, so nobody questions it.
樣品座裡的粉末並不是那個材料,而是材料經過研磨、篩分、填裝、壓平之後變成的東西。The powder in the holder is not the material; it is what the material became after grinding, sieving, packing and pressing.
先看看那邊同學的彈藥:他用環磨機磨了 30 分鐘,說是「確保夠細」。去調查它。First, a classmate's ammo over there: thirty minutes in a ring mill, "to make sure it's fine enough". Go and Use it.
沒錯:尺寸、應變,甚至相本身。所有人為誤差都是「尺寸問題」,可用窗口約 0.2–5 μm。Exactly: size, strain, even the phase itself. Every artefact is a size problem; the usable window is about 0.2–5 μm.
換你了:下一關要定量石膏。石膏是片狀,而且受熱會脫水。到工作台製作彈藥,再到靶場試射。Your turn: next you quantify gypsum. Gypsum is platy and dehydrates when heated. Make ammo at the bench and test it on the range.
好彈藥:低速手磨、加液體、側向裝填。石膏兵來了!Good ammo: gentle hand grinding, a liquid, side-loading. Here come the gypsum soldiers!
太粗:微吸收、消光、參與晶粒太少、擇優取向;太細:0.2 μm 以下峰寬化,過度研磨還有應變。剛好:約 5 μm、側向裝填、旋轉、慢速計數。Too coarse: microabsorption, extinction, too few grains, orientation. Too fine: broadening below 0.2 μm, plus strain. Just right: about 5 μm, side-loaded, spun, counted slowly.
試樣終於誠實了。現在,把它帶到指認會前面。The specimen is honest at last. Now bring it to the line-up.
調查同學的彈藥(環磨機 30 分鐘),找出三個問題。Use the classmate's ammo (ring mill, 30 min) and find three problems.
在工作台製作石膏彈藥並到靶場試射;目標:理想粒徑、不加熱、側向裝填。Make gypsum ammo at the bench and test it; aim for the ideal size, no heating, side-loading.
粒徑窗口:>45 μm 太粗;5–45 μm 偏粗;0.2–5 μm 理想;<0.2 μm 峰寬化。石膏怕熱也怕被壓平。Size window: >45 too coarse, 5–45 coarse, 0.2–5 ideal, <0.2 μm broadened. Gypsum hates heat and pressing.
用你的彈藥擊敗石膏兵。Defeat the gypsum soldiers with your ammo.
同學的筆記如圖。環磨機乾磨 30 分鐘後,圖譜可能出現哪三個問題?(選三個後送出)The classmate's notes are shown. After 30 min dry ring-milling, which three problems may appear? (pick three, then submit)
粒徑過細(<0.2 μm)造成峰寬化Too fine (<0.2 μm): broadened peaks
晶格應變造成峰形扭曲、不對稱Lattice strain: distorted, asymmetric peaks
局部加熱可能讓相轉變(例如石膏脫水)Local heating may transform a phase (e.g. gypsum dehydrates)
微吸收(顆粒太粗)Microabsorption (grains too coarse)
Kβ 假峰Kβ ghosts
低角度溢射到樣品座Low-angle spill onto the holder
再想想:「磨太久」會怎樣?尺寸、應變,還有熱。Think again: what does over-grinding do? Size, strain, and heat.
成品:{c},Sprep = {s}Result: {c}, Sprep = {s}
警告:研磨發熱,石膏可能已脫水成半水石膏!Warning: grinding heat may have dehydrated the gypsum!
正面壓實:片狀石膏的 (020) 會異常強、其他晶面偏弱。Front-pressed: platy gypsum's (020) will be too strong, the rest too weak.
理想彈藥完成!到靶場試射一發。Ideal ammo ready! Test-fire on the range.
石膏的峰不見了——它已經脫水變成別的相!The gypsum peaks are gone: it has dehydrated into another phase!
試射:Sprep = {s},傷害 {d}Test: Sprep = {s}, damage {d}
真相永遠只有一個——本檔案中的每一種人為誤差都是「尺寸問題」,可用的窗口大約落在 0.2 至 5 微米之間。There is only one truth: every artefact in this file is a size problem, and the usable window is roughly 0.2 to 5 micrometres.
熟料在環磨機磨了 30 分鐘,圖譜可能出現的三個問題是?Clinker ring-milled for 30 minutes: the three likely problems?
- 微吸收、擇優取向、KβMicroabsorption, orientation, Kβ
- 零點偏移、試樣位移、螢光Zero offset, displacement, fluorescence
- 過細造成峰寬化、晶格應變使峰形不對稱、局部加熱可能讓相轉變Broadening from over-fine grains, strain asymmetry, heat-driven phase change
- 沒有問題,越細越好None: finer is always better
尺寸、應變,甚至可能是相本身。Size, strain, and possibly the phase itself.
要定量水泥中的石膏,製樣應如何安排?How should you prepare a cement to quantify its gypsum?
- 環磨機乾磨 30 分鐘、正面壓實30 min dry ring mill, front-pressed
- 低速手磨、加液體、側向裝填,避免加熱Gentle hand grinding with a liquid, side-loaded, no heating
- 不研磨直接測No grinding
- 先在 500 °C 烘乾Dry at 500 °C first
石膏呈片狀(怕擇優取向),而且受熱會脫水。Gypsum is platy and dehydrates when heated.
講義建議的可用粒徑窗口是?The usable particle-size window?
- 0.2–5 μm
- 5–45 μm
- >45 μm
- <0.2 μm
研磨至約 5 μm 避免吸收與消光誤差,但須大於 0.2 μm 以防峰寬化。About 5 μm, but above 0.2 μm.
計數統計:要把峰的精度提高一倍,量測時間要?Counting statistics: to double the precision, measurement time must?
- 減半Halve
- 不變Stay the same
- 加倍Double
- 變成四倍(不確定度 ∝ 1/√N)Quadruple (uncertainty ∝ 1/√N)
精度與 √N 成正比。Precision ∝ √N.
Part 03 所說的「消光」與 Part 01 的 F = 0 消光有何不同?How does Part 03 "extinction" differ from the F = 0 extinction of Part 01?
- 完全相同They are identical
- 前者是大而完美的晶體因連續晶面繞射而強度衰減,可把粒徑控制在約 5 μm 以下避免;後者是結構因子為零The former is intensity loss in large, perfect crystals (avoid by grains <~5 μm); the latter is a zero structure factor
- 前者是 Kβ 造成The former is caused by Kβ
- 後者可以靠研磨消除The latter is removed by grinding
一個寫在結構因子裡,另一個寫在你的研缽與杵上。One is in the structure factor; the other is in your mortar and pestle.
水泥廠,旋窯前。玻璃後面站著幾位老面孔:alite、belite、鋁酸鹽、鐵鋁酸鹽,以及一整個硫酸鹽家族。The cement plant, before the kiln. Behind the glass stand the usual suspects: alite, belite, aluminate, ferrite, and the whole sulfate family.
吾乃熟料巨像。來吧,偵探——用你的槍指認我啊!I am the Clinker Colossus. Come, detective: identify me with your gun!
小心:四個主相的最強線全擠在 30–35°,那裡幾乎每一根峰都是兩個以上相的重疊。Careful: the four main phases put their strongest lines in 30–35°; nearly every peak there is an overlap.
單峰峰高無效——任何一槍都同時打到好幾個相。當圖譜擠到讀不出來時,不該瞇更緊的眼睛。Single-peak heights are useless: every shot hits several phases. When a pattern is too crowded to read, don't squint harder.
用化學方法請部分嫌犯出場!SAM 彈(水楊酸–甲醇)溶掉 alite、belite 與游離石灰;KOH–糖彈溶掉鋁酸鹽與鐵鋁酸鹽。Chemically remove some suspects! SAM (salicylic acid–methanol) dissolves alite, belite and free lime; KOH–sugar dissolves aluminate and ferrite.
SAM 殘餘物再「灼燒 500 °C」,趕走石膏的水分,鹼金屬硫酸鹽才會現形。在繞射儀面板選手榴彈種類,再投擲到巨像身上。Ignite the SAM residue at 500 °C to drive off the gypsum water, and the alkali sulfates appear. Pick the grenade type on the panel, then throw it at the Colossus.
所有相都指認了!但巨像正在切換晶型——偽裝戰。看 51.5° 附近的峰形,對照參考圖。Every phase identified! But the Colossus is switching polymorphs: the disguise battle. Read the shape near 51.5° against the references.
……被看穿了。可是,你的槍對我無效。...Seen through. But your gun cannot touch me.
巨像的外殼崩落,露出真身——玻璃相首腦!沒有長程有序,繞射槍只打得出寬鼓包。The shell crumbles, revealing the Glass Phase leader! No long-range order: the gun only raises a broad hump.
還記得第一案嗎?玻璃相在鹼液中會溶解!用鹼液手榴彈!Remember case one? Glass dissolves in alkali! Use the alkali grenades!
玻璃相首腦被揭穿,布拉格偵探社正式結案。The Glass Phase leader is unmasked; the Bragg Detective Agency closes the file.
吸收、製備、指認。三份檔案,一個結論——Absorption, preparation, identification. Three case files, one conclusion:
XRD 給出的數字,永遠不會比它所來自的那份粉末更可靠。A number from an XRD is only as good as the powder it came from.
試著射擊熟料巨像(例如 30–35° 區)。Try firing at the Colossus (e.g. the 30–35° region).
用 SAM/KOH–糖/灼燒簡化圖譜,逐一擊破讀得出來的相(面板選彈種後投擲)。Simplify with SAM / KOH–sugar / ignition and defeat each readable phase (pick the type on the panel, then throw).
SAM 後:方鎂石、C₃A、C₄AF、二水/半水/無水石膏;再灼燒:K₂SO₄。KOH–糖後:alite、belite、CH。相近的峰請用窄狹縫。After SAM: periclase, C₃A, C₄AF, gypsum, hemihydrate, anhydrite; then ignite: K₂SO₄. After KOH–sugar: alite, belite, CH. Use a narrow slit for close peaks.
用鹼液手榴彈擊敗玻璃相首腦!Defeat the Glass Phase leader with alkali grenades!
巨像在 51.5° 附近的峰形(紅)與三張參考圖(示意)如下。alite 是哪一種晶型?The Colossus' profile near 51.5° (red) and three schematic references. Which alite polymorph is it?
T1(三斜)T1 (triclinic)
M1(單斜)M1 (monoclinic)
M3(單斜)M3 (monoclinic)
熟料中驗出 γ-C₂S 與 T1 型 alite。窯爐操作最可能出了什麼問題?A clinker shows γ-C₂S and T1 alite. What most likely went wrong at the kiln?
熟料冷卻太慢:β → γ 轉變(體積膨脹、粉化),alite 也沒被穩定在 M 型Cooling far too slow: β → γ (expansion, dusting), and alite not held in the M form
終粉磨溫度太高Finish grinding too hot
石膏添加不足Not enough gypsum
鹼含量太低Too little alkali
{n} 擊破!{n} defeated!
鹼液生效:玻璃相正在溶解!Alkali works: the glass is dissolving!
SAM 萃取:alite、belite、游離石灰溶解;方鎂石、C₃A、C₄AF 與石膏家族現形!SAM: alite, belite and free lime dissolve; periclase, C₃A, C₄AF and the gypsum family appear!
KOH–糖萃取(另取一份試樣):鋁酸鹽與鐵鋁酸鹽溶解;alite、belite、CH 現形!KOH–sugar (fresh aliquot): aluminate and ferrite dissolve; alite, belite and CH appear!
灼燒 500 °C:水分趕走,鹼金屬硫酸鹽 K₂SO₄ 現形!Ignition at 500 °C: water gone, the alkali sulfate K₂SO₄ appears!
灼燒要用在 SAM 殘餘物上:請先投 SAM。Ignite the SAM residue: throw SAM first.
二水與半水石膏脫水,已確認身分。Gypsum and hemihydrate dehydrated and confirmed.
判斷錯誤,巨像反擊!Wrong: the Colossus strikes back!
真相永遠只有一個——當圖譜擠到讀不出來時,你不該瞇更緊的眼睛,而該用化學方法把部分嫌犯請出場。There is only one truth: when a pattern is too crowded to read, you do not squint harder; you chemically remove some of the suspects.
十二份檔案,一套方法。XRD 給出的數字,永遠不會比它所來自的那份粉末更可靠。本課程接下來將離開繞射儀、改拿電子束——SEM 與 X 光微區分析。感謝遊玩,偵探!Twelve case files, one method. A number from an XRD is only as good as the powder it came from. Next in this course: the electron beam, SEM and X-ray microanalysis. Thanks for playing, detective!
為什麼普通水泥不能用 30–35° 的單峰峰高定量?Why can't single-peak heights at 30–35° quantify a cement?
- alite、belite、C₃A、C₄AF 的最強線都擠在那裡,幾乎每根峰都是重疊The strongest lines of alite, belite, C₃A and C₄AF all overlap there
- 那裡沒有峰There are no peaks there
- 那裡只有 KβOnly Kβ there
- 儀器量不到那個角度The instrument can't reach it
所以水泥定量需要全圖譜擬合與化學萃取。Hence whole-pattern fitting and extractions.
SAM(水楊酸–甲醇)萃取會移除哪些相?What does a SAM extraction remove?
- 鋁酸鹽與鐵鋁酸鹽Aluminate and ferrite
- 所有硫酸鹽All sulfates
- alite、belite 與游離石灰Alite, belite and free lime
- 方鎂石Periclase
殘餘物可標定方鎂石、C₃A、C₄AF、二水/半水/無水石膏與 K₂SO₄。The residue shows periclase, C₃A, C₄AF, the calcium sulfates and K₂SO₄.
KOH–糖萃取會移除哪些相?What does a KOH–sugar extraction remove?
- alite 與 beliteAlite and belite
- 鋁酸鹽與鐵鋁酸鹽Aluminate and ferrite
- CHCH
- 全部Everything
留下 alite、belite 與 CH;兩種互補萃取合起來交代整份水泥。It leaves alite, belite and CH; together the two extractions cover the cement.
出廠水泥驗出 γ-C₂S,代表什麼?Finding γ-C₂S in a delivered cement means?
- 燒成溫度太高Burning too hot
- 石膏太多Too much gypsum
- 水化太快Hydration too fast
- 熟料冷卻太慢:β → γ 轉變伴隨體積膨脹粉化,γ 幾乎沒有水硬性The clinker cooled too slowly: β → γ expands and dusts, and γ is nearly inert
急速冷卻與微量元素可把 belite 留在 β 型。Rapid cooling and stabilisers keep belite in the β form.
要判斷水泥廠是否「粉磨過熱」,SAM 殘餘圖譜該看什麼?To judge whether a plant over-heats during grinding, what do you read in the SAM residue?
- 二水石膏、半水石膏、無水石膏的比例(跟著水走)The gypsum : hemihydrate : anhydrite ratio (follow the water)
- alite 的強度Alite intensity
- 方解石的峰位Calcite position
- Kβ 的數量The number of Kβ lines
三者比例取決於終粉磨時水泥有多熱。Their proportions record how hot the finish grinding got.
alite 的晶型(T1、M1、M3)主要由哪裡判斷?How is the alite polymorph (T1, M1, M3) identified?
- 32° 最強峰的高度Height of the 32° peak
- 顏色Colour
- 51.5° 附近的峰形The peak shape near 51.5°
- 鋁酸鹽的 33.3° 峰The aluminate peak at 33.3°
晶型記錄窯的燒成與冷卻條件,也影響早期水化速率;混合晶型永遠可能。The polymorph records the kiln history; mixtures are always possible.