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晶体学·固体结构
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晶体学·固体结构
12
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布拉格衍射 · nλ = 2d·sinθ
Bragg diffraction · nλ = 2d·sinθ
nλ=2d·sinθ · 晶面相长反射 · 程差2d·sinθ
When two X-rays reflected from crystal planes interfere constructively
晶面指数 (hkl)
Miller indices (hkl)
截距倒数化整 · (100)/(110)/(111)/(210)立方晶面
Naming a crystal plane by the reciprocals of its axis intercepts, cleared of fractions
7 大晶系与 14 种布拉维格子
The 7 crystal systems & 14 Bravais lattices
7晶系14布拉伐点阵 · P/I/F/C心 · 参数约束
Crystal systems set by lattice-parameter constraints, then centering (P/I/F/C) gives the 14 Bravais lattices
立方晶系:堆积与配位数
Cubic lattices: packing & coordination
SC/BCC/FCC · CN6/8/12 · APF0.52/0.68/0.74 · r-a关系
Comparing SC / BCC / FCC on atoms-per-cell, coordination, packing factor and the r–a relation
六方最密堆积(HCP)
Hexagonal close packing (HCP)
六方密堆 · c/a=√(8/3)≈1.633 · CN12 · APF0.74
Conventional hexagonal cell: 6 atoms, CN 12, ideal c/a = √(8/3), APF 0.7405, ABAB stacking
倒易点阵 · b₁ = 2π(a₂×a₃)/V
Reciprocal lattice · b₁ = 2π(a₂×a₃)/V
b=2π(a×a)/V · 立方倒易仍立方2π/a · |G|=2π/d
The reciprocal of a cubic direct lattice is cubic with parameter 2π/a; |G_hkl| = 2π/d_hkl
立方晶系面间距 · d = a / √(h²+k²+l²)
Cubic interplanar spacing · d = a / √(h²+k²+l²)
立方d_hkl=a/√(h²+k²+l²) · hkl与a定间距
How the (hkl) indices and lattice parameter a set the plane spacing
粉末 X 射线衍射图谱(立方晶系)
Powder X-ray diffraction pattern (cubic)
2θ峰位 · SC全/BCC h+k+l偶/FCC同奇偶允许反射
2θ peak positions from Bragg's law and cubic d-spacing, with the allowed (hkl) reflections set by lattice type
几何结构因子与系统消光
Structure factor & systematic absences
F=Σfⱼe^{2πi(hxⱼ+kyⱼ+lzⱼ)} · BCC/FCC系统消光 · |F|²
F(hkl) = Σ fⱼ·exp[2πi(hxⱼ+kyⱼ+lzⱼ)] — the basis geometry decides which reflections vanish
最密堆积的堆垛序列(HCP vs FCC)
Close-packed stacking (HCP vs FCC)
ABAB=HCP vs ABCABC=FCC · 第三层区分 · 均0.74
Identical close-packed layers; the third layer decides: ABAB… = HCP, ABCABC… = FCC
劳厄条件 · 埃瓦尔德球
Laue condition · Ewald sphere
Δk=G · 埃瓦尔德球构造 · 倒易点落球出衍射
Δk = G: diffraction when the scattering vector equals a reciprocal-lattice vector
晶体理论密度 ρ = nA/(V_c·N_A)
Theoretical crystal density ρ = nA/(V_c·N_A)
ρ=n·A/(V_c·N_A) · Al FCC→2.70 g/cm³
Compute a metal's theoretical (X-ray) density from its unit cell: SC / BCC / FCC worked examples