二、布拉格方程 1. 晶体中原子面的衍射 原子面上任意两点M、M1，平行X光投射到该面， M、M1两点在原子面反射方向上的光程差： δ＝OM－O1M1= MM1 cosθ－ MM1 cosθ＝0 说明该原子面所有原子散射波在反射方向上相位均相同，发生干涉，称该面对入射X射线衍射。 同一晶面上原子的散射线在原子面反射线方向上可以互相加强，形成衍射波。 X射线波长短，穿透力强，晶体内部和表面的原子都成为散射波源，衍射线应被看成是许多平行原子面反射的反射波叠加的结果。 2. 布拉格方程 入射X射线波长为λ，入射角为θ，平行原子面间距为d，任选两面A、B，其法线与原子面交于M、M2。经A、B两面反射的反射波光程差为： δ＝PM2＋ QM2 ＝2dsinθ 干涉加强的条件为：反射方向的散射波相位差为2π整数倍或光程差为波长的整数倍（δ=nλ），可得2dsinθ = nλ . n——任意整数，称为反射级数 d——(hkl)晶面间距 布拉格实验得到了“选择反射”的结果，即当X射线以某些角度入射时，记录到反射线，其他角度入射时，则无反射。 布拉格实验是X射线衍射仪的原型。入射线与反射面的夹角为θ，称为掠射角或布拉格角。 记录装置与样品台以2∶1的角速度同步转动，保证记录装置始终处于接收反射线的位置上。 ---**Image Description:** * **Type:** Ray diagram illustrating reflection of light rays from a horizontal surface. * **Base Line:** A horizontal blue line, labeled 'A' at both ends. Several blue points are located on this line, including two specifically labeled points M1 and M. * **Rays:** * A red ray, labeled 'L1', originates from the upper left, is incident on the blue line at point M1, and reflects upwards and to the right as a red ray labeled 'N1'. Both L1 and N1 have arrows indicating direction. * A green ray, labeled 'L', originates from the upper left, is incident on the blue line at point M, and reflects upwards and to the right as a green ray labeled 'N'. Both L and N have arrows indicating direction. * **Normals:** * A black dashed line is drawn perpendicular to the blue line at M1, extending upwards. A right-angle symbol indicates perpendicularity. This normal intersects the ray L1 at point 'o'. * A black dashed line is drawn perpendicular to the blue line at M, extending upwards. A right-angle symbol indicates perpendicularity. This normal intersects the ray L at point 'o1'. * **Angles:** An angle labeled 'θ' is shown between the incident ray L1 and the horizontal blue line. * **Other Elements:** * A diagonal line is shown on the upper left, parallel to the direction of L1 and N1, labeled with Chinese characters "同位向" (Tóngwèi xiàng), which translates to "Same Direction" or "Homotopic Direction". * A diagonal line is shown on the upper right, parallel to the direction of L and N, labeled with Chinese characters "同光程" (Tóng guāngchéng), which translates to "Same Optical Path" or "Equidistant". **Extracted Text:** * 同位向 * 同光程 * L * L1 * N1 * N * θ * o * o1 * A * M1 * M **Textual Information:** * 同位向 (Homologous Direction) * 同光程? (Same optical path length?) * Labels: A, B, d, θ, L, L1, M, M1, M2, N, N1, N2, P, Q **Diagram Description:** * **Type:** Geometric diagram illustrating reflection from two parallel lines representing planes or layers. * **Main Elements:** * Two parallel horizontal lines labeled A and B. Line A is above line B. * The vertical distance between lines A and B is labeled d. * Multiple blue dots are shown on lines A and B, representing points on the planes. Specific points labeled are M and M1 on line A, and M2, P, and Q on line B. * An incident ray L (green line with arrow) strikes line A at point M and reflects along ray N (green line with arrow). The angle between the incident ray L and line A is labeled θ. The angle between the reflected ray N and line A is labeled θ. * An incident ray L1 (red line with arrow) strikes line A at point M1, then travels along the segment M1M2 (red line) to point M2 on line B. At M2, it reflects along ray N2 (red line with arrow). The angle between the segment M1M2 and line B is labeled θ. The angle between the reflected ray N2 and line B is labeled θ. An additional reflected ray N1 (red line with arrow) is shown reflecting from M1 on line A. * The incident rays L and L1 are annotated with "同位向" (Homologous Direction). * The reflected rays N, N1, N2 are annotated with "同光程?" (Same optical path length?). * Segments MP and MQ are drawn from point M on line A to points P and Q on line B. Right angle symbols at P and Q indicate that MP is perpendicular to line B and MQ is perpendicular to line B. The length of MP and MQ is labeled d. * Triangle MMP is shown, with a right angle at P and an angle labeled θ at M2. The vertices of this triangle are M, P, and M2. Similarly, triangle MMQ is shown with a right angle at Q and an angle labeled θ at M2. The vertices are M, Q, and M2. (Note: The vertex labeled Q is the same position as M2 in the diagram, and the angle θ is indicated at this position in both triangles). The labels P and Q are specifically placed at the feet of the perpendiculars from M to line B. * **Implied Relationships:** The diagram illustrates the reflection of waves from two parallel planes A and B. The angles of incidence (with respect to the plane) equal the angles of reflection (with respect to the plane) for both reflections shown. The presence of two ray paths (reflection from A only, and reflection from A then B then back) and the annotation "同光程?" (Same optical path length?) suggest a comparison of optical path lengths, likely related to interference or diffraction phenomena like Bragg's Law. The distance d is the spacing between the planes. The angle θ is the grazing angle. The segments MP and MQ illustrate the perpendicular distance d between the planes. The triangles MMP and MMQ with angle θ at M2 and a right angle at P/Q relate the distance MM2 to d and θ. Based on standard Bragg's law geometry, the path difference between the ray reflecting from A at M and the ray reflecting from B at M2 (after initial interaction at M1 on A) is related to d and θ. The angles labeled θ within the triangles at M2, along with the perpendiculars from M to B of length d, are likely intended to facilitate the calculation of this path difference. In right triangle MMP, sin(θ) = MP/MM2 = d/MM2. The path difference is typically 2d sin(θ). **Diagram Description:** * **Type:** Geometric diagram illustrating reflection/diffraction of waves from a surface, likely related to Bragg's Law. * **Main Elements:** * A horizontal line labeled 'A' represents a reflecting surface or a plane of atoms. * Two points, 'M₁' and 'M', are located on line 'A'. * An incoming ray labeled 'L₁' (red line with arrow) is incident on point 'M₁'. * A reflected ray labeled 'N₁' (red line with arrow) originates from 'M₁'. * An incoming ray labeled 'L' (green line with arrow) is incident on point 'M'. * A reflected ray labeled 'N' (green line with arrow) originates from 'M'. * Two perpendicular lines (black lines) are drawn from M₁ to the incoming ray L₁ and from M to the incoming ray L, indicated by right-angle symbols. * Two parallel diagonal lines are present: one labeled "同位向" (Same Direction) above L₁, and one labeled "同光程" (Same Optical Path) above N₁. * An angle $\theta$ is shown between the incoming ray 'L' and the horizontal line 'A'. This angle is also shown between the reflected ray 'N' and the horizontal line 'A'. * A dashed line extends the path of the incoming ray 'L'. * An angle $2\theta$ is shown between the extended incoming ray 'L' (dashed line) and the reflected ray 'N'. A curved arrow indicates this angle. * **Labels and Annotations:** * "同位向" (Same Direction) is associated with the direction of incoming rays L₁ and L. * "L₁", "L", "N₁", "N" are labels for the rays. * "A", "M₁", "M" are labels for points/line. * "布拉格角 (掠射角)" (Bragg Angle (Grazing Angle)) points to the angle $\theta$. * "同光程" (Same Optical Path) is associated with the reflected ray N₁. * "衍射角" (Diffraction Angle) points to the angle $2\theta$. * $\theta$ is labeled next to the angle between L and A, and N and A. * $2\theta$ is labeled next to the angle between the extended L and N. **Textual Information:** * **Labels:** 同位向, L, L₁, A, M₁, M, 布拉格角 (掠射角), N₁, N, 同光程, 衍射角. * **Angles:** $\theta$, $2\theta$. * **Annotation for $\theta$:** 布拉格角 (掠射角) (Bragg Angle (Grazing Angle)). * **Annotation for $2\theta$:** 衍射角 (Diffraction Angle).