Miller indices and Diffraction

# Miller indices length In a cubic crystal system, the Miller indices $(hkl)$ denote lattice planes, and the length associated with these planes is usually interpreted as the interplanar spacing, $d_{hkl}$. For a cubic unit cell with lattice constant $a$, the interplanar distance $d_{hkl}$ is given by the formula: $$ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} $$ Here, $a$ is the cube edge length, and $(hkl)$ are the Miller indices for the plane in question. This relationship works for simple cubic, face-centered cubic, and body-centered cubic lattices when $a$ is referenced to the cubic supercell axes.[^1_1][^1_4] ## Key Points - Miller indices $(hkl)$ specify crystal planes based on fractional intercepts with the axes, followed by taking reciprocals and reducing to the smallest integers.[^1_2][^1_6] - In cubic systems, all lattice vectors are equal in length ($a$), so calculation is straightforward.[^1_4][^1_1] - The physical meaning of the Miller indices is visualized by the orientation (normal) and spacing of the planes in the 3D lattice, where higher $h$, $k$, or $l$ values correspond to planes closer together.[^1_1][^1_2] ## Example Table | Miller indices (hkl) | Plane intercepts (in $a$) | Interplanar spacing ($d_{hkl}$) | | :-- | :-- | :-- | | (100) | $a, \infty, \infty$ | $a/1 = a$ | | (110) | $a, a, \infty$ | $a/\sqrt{2}$ | | (111) | $a, a, a$ | $a/\sqrt{3}$ | These calculations and this method are central to the interpretation of X-ray diffraction and the geometric analysis of crystal structures.[^1_2][^1_4][^1_1] [^1_3][^1_5][^1_7][^1_8][^1_9] [^1_1]: https://en.wikipedia.org/wiki/Miller_index [^1_2]: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Surface_Science_(Nix)/01:_Structure_of_Solid_Surfaces/1.02:_Miller_Indices_(hkl) [^1_3]: http://lampz.tugraz.at/~hadley/ss1/crystalstructure/miller.php [^1_4]: https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/06:_Recitations/6.14:_Miller_Indices_and_Interplanar_Spacing [^1_5]: https://www.uobabylon.edu.iq/eprints/publication_5_28480_84.pdf [^1_6]: https://uomustansiriyah.edu.iq/media/lectures/6/6_2021_02_16!07_46_44_PM.pdf [^1_7]: https://www.youtube.com/watch?v=_4gye7kd1Xg [^1_8]: https://ocw.nthu.edu.tw/ocw/upload/114/news/CHAPTER02 revised 1.pdf [^1_9]: http://www.cityu.edu.hk/phy/appkchu/ap5301/Electron Diffraction - Paper.pdf --- To prove the cubic lattice formula for interplanar spacing, $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$, begin with the geometry of the planes described by Miller indices $(hkl)$: 1. For a cubic crystal, the intercepts of a plane with Miller indices $(hkl)$ along the axes are $a/h, a/k, a/l$, where $a$ is the cubic lattice parameter.[^2_1] 2. The perpendicular distance ($d_{hkl}$) between successive parallel planes is the interplanar spacing. Using direction cosines and the properties of orthogonal axes: $$ \cos \alpha = \frac{d_{hkl}}{a/h}, \quad \cos \beta = \frac{d_{hkl}}{a/k}, \quad \cos \gamma = \frac{d_{hkl}}{a/l} $$ where $\alpha, \beta, \gamma$ are the angles made with the axes.[^2_1] 3. For orthogonal axes, the sum of square cosines is 1: $$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $$ 4. Substitute the expressions for the cosines: $$ \left(\frac{d_{hkl}}{a/h}\right)^2 + \left(\frac{d_{hkl}}{a/k}\right)^2 + \left(\frac{d_{hkl}}{a/l}\right)^2 = 1 $$ $$ \left(\frac{d_{hkl}^2}{a^2/h^2}\right) + \left(\frac{d_{hkl}^2}{a^2/k^2}\right) + \left(\frac{d_{hkl}^2}{a^2/l^2}\right) = 1 $$ $$ d_{hkl}^2 \left( \frac{h^2 + k^2 + l^2}{a^2} \right) = 1 $$ 5. Solving for $d_{hkl}$: $$ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} $$ This derivation is a standard result for simple cubic systems, stemming from the relationship between the plane orientation and the lattice axes.[^2_2][^2_1] [^2_3][^2_4][^2_5][^2_6] [^2_1]: https://phy.buet.ac.bd/faculty_profile/files/1029553953.pdf [^2_2]: https://en.wikipedia.org/wiki/Miller_index [^2_3]: https://www.youtube.com/watch?v=mKBdq99M1hg [^2_4]: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Surface_Science_(Nix)/01:_Structure_of_Solid_Surfaces/1.02:_Miller_Indices_(hkl) [^2_5]: https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/06:_Recitations/6.14:_Miller_Indices_and_Interplanar_Spacing [^2_6]: https://web.pdx.edu/~pmoeck/phy381/lecture3.pdf ## How to get this equation ? Let's break down the proof for the interplanar spacing in a cubic lattice by considering each axis: ### **Step 1: Find Plane Intercepts** For a cubic lattice with unit cell length $a$, the intercepts of the $(hkl)$ plane are: - $x$-axis: $a/h$ - $y$-axis: $a/k$ - $z$-axis: $a/l$ ### **Step 2: Equation of the Plane** The general equation for such a plane is: $$ \frac{x}{a/h} + \frac{y}{a/k} + \frac{z}{a/l} = 1 $$ Multiply both sides by $a$ to get: $$ h \cdot x + k\cdot y + l\cdot z = a $$ ### **Step 3: Find the Normal Vector** The vector normal (perpendicular) to the $(hkl)$ plane is: $$ \vec{n} = h\cdot \hat{x} + k\cdot \hat{y} + l\cdot \hat{z} $$ The length of this normal vector is $$ \sqrt{h^2 + k^2 + l^2} $$ ### **Step 4: Spacing Between Parallel Planes** The spacing $d_{hkl}$ is the perpendicular distance between two adjacent $(hkl)$ planes. The general formula for the distance from the origin to the plane $h\cdot x + k\cdot y + l\cdot z = a$ is: $$ d_{hkl} = \frac{|a|}{\sqrt{h^2 + k^2 + l^2}} $$ ### **Summary Table** | Step | Detail | | :---- | :---- | | Intercepts | $a/h, a/k, a/l$ | | Plane eqation | $h\cdot x + k\cdot y + l\cdot z = a$ | | Normal | $(h, k, l)$ | | Spacing | $d_{hkl}= \frac{a}{\sqrt{h^2+ k^2+ l^2}}$ | **This method separates the argument into each axis (x, y, z), finds the intercepts, constructs the general plane equation, identifies the normal, and uses distance-from-origin-to-plane to get the standard spacing formula.** # Miller-Bravais Indices The Miller-Bravais Indices are an extension of Miller Indices, specifically developed to describe directions and planes in the hexagonal crystal system, where the geometry involves four axes (three in the basal plane, 120° apart, and one perpendicular). ## Why Miller-Bravais Indices? - The hexagonal lattice has symmetry that isn't captured fully by standard three-index (Miller) notation. - To reflect the hexagonal symmetry, Miller-Bravais notation uses four indices: $(h k i l)$ for planes and $[u v t w]$ for directions. - The third index, $i$ (for planes) or $t$ (for directions), is redundant: $i = - (h + k)$ and $t = - (u + v)$. ### Notation for Planes: - Miller: $(hkl)$ - Miller-Bravais (hexagonal): $(h k i l)$ - $i = -(h + k)$ - Example: The $(1 0 \overline{1} 0)$ plane ### Notation for Directions: - Miller: $[UVW]$ ### UVW directions to uvtw directions To convert direction indices in the hexagonal system from 3-index Miller notation $[UVW]$ to 4-index Miller-Bravais notation $[uvtw]$, use the following formulas: $$ u = \frac{1}{3}(2U - V) $$ $$ v = \frac{1}{3}(2V - U) $$ $$ t = - (u + v) = - (U + V) $$ $$ w = W $$ - Here, $[UVW]$ are the 3-index coordinates, and $[uvtw]$ are the 4-index Miller-Bravais coordinates. - The index $t$ is redundant and equals $- (u + v)$. This conversion captures the symmetry of the hexagonal lattice and properly expresses directions in hexagonal close-packed crystals. - Miller-Bravais (hexagonal): $[u v t w]$ - $t = -(u + v)$ ### Applications: - Used for hexagonal close-packed (HCP) crystal structures (e.g., in magnesium, titanium, zinc, etc.). - Makes it easier to identify symmetry-related planes and directions. - Essential for proper description of slip directions, growth facets, and surface orientations in hexagonal crystals. #### Summary Table | System | Planes | Directions | Relations | | :-- | :-- | :-- | :-- | | Cubic | $(hkl)$ | $[uvw]$ | n/a | | Hexagonal | $(hkil)$ | $[uvtw]$ | $i = -(h+k)$, $t = -(u+v)$ | The Miller-Bravais system allows for a clearer, symmetric representation of geometry in crystals with hexagonal symmetry. [^3_1][^3_10][^3_11][^3_12][^3_13][^3_14][^3_2][^3_3][^3_4][^3_5][^3_6][^3_7][^3_8][^3_9] [^3_1]: https://www.ucl.ac.uk/~ucapahh/teaching/3C25/Lecture03s.pdf [^3_2]: https://www.youtube.com/watch?v=eDS2rWQu1TY [^3_3]: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Surface_Science_(Nix)/01:_Structure_of_Solid_Surfaces/1.02:_Miller_Indices_(hkl) [^3_4]: https://ocw.nthu.edu.tw/ocw/upload/114/news/CHAPTER02 revised 1.pdf [^3_5]: https://phy.buet.ac.bd/faculty_profile/files/1029553953.pdf [^3_6]: https://www.youtube.com/watch?v=QbP6J25IEAI [^3_7]: https://web.stanford.edu/group/glam/xlab/MatSci162_172/LectureNotes/02_Geometry,%20RecLattice.pdf [^3_8]: https://en.wikipedia.org/wiki/Miller_index [^3_9]: https://www.physics.muni.cz/~jancely/teamCMV/Texty/RuzneTexty/Krystaly/CrystalPlanes\&Diffraction.pdf [^3_10]: https://www.uobabylon.edu.iq/eprints/publication_5_28480_84.pdf [^3_11]: http://www.phys.nchu.edu.tw/upload/download_files/a5027620faf2cc0b7e254ee2a554a8fe.pdf [^3_12]: https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/06:_Recitations/6.14:_Miller_Indices_and_Interplanar_Spacing [^3_13]: https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781119961468.app3 [^3_14]: https://fiveable.me/condensed-matter-physics/unit-1/miller-indices/study-guide/oSNWdfZMEoB7gifb # X-Ray Diffraction and Crystal Structure Determination ## What is X-Ray Diffraction? X-ray diffraction (XRD) refers to the scattering of X-rays when they encounter the regularly spaced planes of atoms in a crystal lattice. Because X-rays have wavelengths similar to atomic spacings, their interaction with the crystal causes constructive and destructive interference, leading to a pattern of diffracted beams.[^1][^2] **Bragg’s Law** governs this phenomenon: $$ n\lambda = 2d \sin\theta $$ - $n$: integer (order of reflection) - $\lambda$: X-ray wavelength - $d$: distance between crystal planes - $\theta$: angle of incidence/reflection <img src=https://cdn.britannica.com/28/6828-050-211C30F8/Bragg-diffraction.jpg> ## How to Use X-Ray Diffraction to Determine a Crystal Structure 1. **X-Ray Source and Sample Placement:** Direct a monochromatic X-ray beam at a crystalline sample. 2. **X-Ray Scattering:** The X-rays interact with the atoms inside the crystal. Only specific angles satisfy Bragg’s law and yield constructive interference.[^3][^4] 3. **Detection:** Detectors (or photographic film) record the intensity and position of the diffracted beams. The result is a diffraction pattern of spots or rings.[^3] 4. **Analysis:** The angles and intensities of these spots are measured. By applying Bragg’s law, the spacings between atomic planes can be determined, revealing the crystal’s lattice parameters and helping reconstruct the atomic arrangement in three dimensions.[^2][^4] ![image](https://hackmd.io/_uploads/HyPKqrmall.png) <img src=https://myscope.training/images/XRD-module/figure_28.jpg> [^1]: https://en.wikipedia.org/wiki/X-ray_diffraction [^2]: https://www.britannica.com/science/X-ray-diffraction [^3]: https://www.cif.iastate.edu/services/acide/xrd-tutorial/xrd [^4]: https://en.wikipedia.org/wiki/X-ray_crystallography [^5]: https://www.youtube.com/watch?v=QHMzFUo0NL8 [^6]: https://web.pdx.edu/~pmoeck/phy381/Topic5a-XRD.pdf [^7]: https://chem.libretexts.org/Courses/Franklin_and_Marshall_College/Introduction_to_Materials_Characterization__CHM_412_Collaborative_Text/Diffraction_Techniques/X-ray_diffraction_(XRD)_basics_and_application [^8]: https://rigaku.com/hubfs/2024 Rigaku Global Site/Resource Hub/Knowledge Library/Rigaku Journals/Volume 37(1) - Winter 2021/Rigaku Journal 37-1_12-19.pdf?hsLang=en [^9]: https://myscope.training/XRD_XRD_basics [^10]: https://www.ebsco.com/research-starters/science/determining-crystal-structures [^11]: https://ywcmatsci.yale.edu/sites/default/files/files/s10832-021-00263-6.pdf [^12]: https://www.esrf.fr/home/UsersAndScience/Experiments/StructMaterials/ID19/Techniques/Diffraction/Overview.html [^13]: https://www.tcd.ie/nanoscience/assets/documents/experiments/Instruction-X-ray Diffraction.pdf [^14]: https://www.pulstec.net/single-crystal-xrd-versus-powder-xrd/ [^15]: https://www.nature.com/articles/s41524-023-01096-3 [^16]: https://www.esrf.fr/home/news/tech-talk/content-news/tech-talk/ultra-compact-x-ray-diffraction-setup-for-sample-characterization-at-id20.html [^17]: https://tpsbl.nsrrc.org.tw/userdata/upload/21A/CDR.pdf [^18]: https://forcetechnology.com/en/articles/identification-crystalline-materials-x-ray-diffraction [^19]: https://chem.libretexts.org/Courses/Rutgers_University/Chem_160:_General_Chemistry/13:_Phase_Diagrams_and_Crystalline_Solids/13.03:_Crystalline_Solids-_Determining_Their_Structure_by_X-Ray_Crystallography [^20]: https://fenix.ciencias.ulisboa.pt/downloadFile/2251937252644078/RaiosX_parte1_20202021_v2_english.pdf