CT5812701 · Case Files #5–#6 · Where the peaks are, and which ones survive
Bragg's Law & Structure Factor布拉格定律與結構因子
Two linked tools: where a peak can appear, and whether it actually does. 兩個相連的工具：峰可以出現在哪裡，以及它究竟會不會出現。
Drag the incident beam to change θ · or use the sliders 拖曳入射光束改變 θ，或使用下方滑桿
drag the beam · 拖曳光束
λ Wavelength 波長0.15418 nm
d Plane spacing 面間距0.3035 nm
θ Bragg angle 布拉格角14.71°
2θ (as plotted 圖譜橫軸)29.42°
2d sin θ0.15408 nm
λ0.15418 nm
Path difference 路徑差
Δ / λ1.00
Δ / λ1.00
In phase — n = 1
同相，產生建設性干涉
Superposition 波的疊加
The two reflected waves and their sum. They reinforce only when the path difference is a
whole number of wavelengths. 兩束反射波與其合成波：唯有路徑差為整數個波長時才會互相加強。
Where the peaks appear 峰出現的位置
Click the chart to jump to a peak. Peak positions are exact; heights are schematic.
點擊圖表可跳到該峰。峰位為精確計算，峰高僅為示意。
Watch the convention 注意定義： in Bragg's law θ is measured from the lattice plane, not from the
normal as in optics — so the angle plotted on a diffractogram is 2θ, the angle between the incident and the
diffracted beam. 布拉格定律中的 θ 是由晶面量起，而非光學慣用的法線 —— 因此繞射圖譜橫軸的 2θ，是入射線與繞射線之間的夾角。
Why λ must be small 為何波長必須夠短： sin θ ≤ 1 forces λ ≤ 2d. With d ≈ 0.3 nm, visible light (λ ≈ 500 nm) can never diffract — only X-rays are the right size. sin θ ≤ 1 迫使 λ ≤ 2d；當 d ≈ 0.3 nm 時， 可見光（λ ≈ 500 nm）永遠無法產生繞射，只有 X 光的尺寸剛好。
Why λ must be small 為何波長必須夠短： sin θ ≤ 1 forces λ ≤ 2d. With d ≈ 0.3 nm, visible light (λ ≈ 500 nm) can never diffract — only X-rays are the right size. sin θ ≤ 1 迫使 λ ≤ 2d；當 d ≈ 0.3 nm 時， 可見光（λ ≈ 500 nm）永遠無法產生繞射，只有 X 光的尺寸剛好。
Bragg's law says a peak may appear. The structure factor decides whether it does. 布拉格定律說峰「可以」出現；結構因子決定它「是否真的」出現。
F(hkl) = Σ fn e2πi(hun+kvn+lwn) — every atom in the cell, added as a vector 晶胞內每顆原子，以向量相加
The arrows are chained tip-to-tail. The red resultant is F(hkl); when they close back on
the origin, F = 0 and the reflection is extinct. 各原子向量首尾相接，紅色合向量即為 F(hkl)；
當向量繞回原點時 F = 0，該反射消光。
Structure 結構
h k l Reflection 反射(1 1 1)
a Lattice parameter 晶格常數0.3615 nm
λ Wavelength 波長0.15418 nm
F(hkl)116.0
Σ f (maximum 上限)116
|F| / Σf
Amplitude 振幅比1.00
Amplitude 振幅比1.00
Allowed
容許反射
Extinction rule 消光規則
The resulting pattern 由此產生的圖譜
Navy peaks are allowed; red dashed ticks mark reflections that Bragg's law permits but
the structure factor wipes out. Click any marker to load it above. Intensities use I ∝ |F|²·p·LP.
深藍為容許反射；紅色虛線標示布拉格定律容許、但被結構因子抹除的反射。點擊任一標記可載入上方分析。
強度採用 I ∝ |F|²·p·LP 計算。
Two kinds of "extinction" 兩種「消光」： this tab shows systematic extinction — F(hkl) = 0 because the
atoms inside the cell cancel one another. That is not the primary extinction of specimen preparation,
which merely weakens a peak that does exist. 本頁展示的是系統性消光：因晶胞內原子互相抵消而 F(hkl) = 0；
這與試樣製備中的一次消光不同，後者只是削弱一個確實存在的峰。
Same geometry, different answer 同樣幾何、不同答案： compare BCC with CsCl. Both have an atom at the body centre, but in CsCl the two atoms are different elements, so the odd reflections no longer cancel and survive as weak superlattice lines. 比較 BCC 與 CsCl：兩者體心都有一顆原子，但 CsCl 的兩種原子不同， 奇數反射不再相消，而以微弱的超晶格譜線存在。
Same geometry, different answer 同樣幾何、不同答案： compare BCC with CsCl. Both have an atom at the body centre, but in CsCl the two atoms are different elements, so the odd reflections no longer cancel and survive as weak superlattice lines. 比較 BCC 與 CsCl：兩者體心都有一顆原子，但 CsCl 的兩種原子不同， 奇數反射不再相消，而以微弱的超晶格譜線存在。
AEML · Advanced Engineering Materials Laboratory, NTUST