Reciprocal Lattice

Reciprocal Lattice Must Read All Hammond - Chapter 6, A5 Krawitz - Chapter 2.8, 2.9, 2.10 Sherwood & Cooper - Chapter 8.7 (page 269 ~ 274; < 6 pages) Cullity - Chapter 2-4, A1-1, A1-2, A1-3 Ott – Chapter 13.3. https://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/index.php https://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Chan Park, MSE-SNU. 1. Intro to Crystallography, 2021. [uvw] & (hkl). [uvw]. (1) a lattice line through the origin and point uvw (2) the infinite set of lattice lines which are parallel to it and have the same lattice parameter. (hkl). an infinite set of parallel planes with a constant interplanar spacing an infinite set of parallel planes which are apart from each other by the same distance interplanar spacing dhkl. Chan Park, MSE-SNU. Intro to Crystallography, 2021. 2. Reciprocal lattice 1/d1. Arbitrary origin. Reciprocal lattice point 1/d2. the direction of vector. the magnitude of vector. direction  orientation of (hkl) magntiude  interplanar spacing. each point in the reciprocal lattice represents a set of planes.  Used to understand  the information in a diffraction pattern  many useful geometric calculations to be performed in crystallography  geometry of X-ray and electron diffraction  behavior of electrons in crystals  Basic concept & application of reciprocal lattice to the analysis of XRD pattern  P. P. Ewald Chan Park, MSE-SNU. Hammond Chap 6. Intro to Crystallography, 2021. 3. 2D reciprocal lattice  For every real lattice, there is an equivalent reciprocal lattice (RL).  A 2-D real lattice is defined by two unit cell vectors, ⃗ and. inclined at an angle γ..  The equivalent RL in reciprocal space is defined by two RL vectors,  magnitude of. ∗ ∗. and. ∗.. = 1/d10 and. ∗⊥. (d10 = interplanar spacing of (10) planes)  magnitude of. ∗. = 1/d01 and. ∗⊥. ⃗. (d01 = interplanar spacing of (01) planes)  A RL can be built using RL vectors.  Both the real and reciprocal constructions show the same lattice, using different but equivalent descriptions. www4.hcmut.edu.vn/~huynhqlinh/project/Minhhoa3/Nhieuxa/Nx2/www.matter.org.uk/diffraction/geometry/ 2D_reciprocal_lattices.htm Chan Park, MSE-SNU. Intro to Crystallography, 2021. 4. 3D reciprocal lattice  In a crystal of any structure, RL vector ∗. is ⊥ to (hkl) plane and has a. length inversely proportional to dhkl ..  Why do we need the concept of reciprocal lattice?  A family of planes can be represented by just one point, which obviously simplifies things.  It offers us a very simple geometric model that can interpret the diffraction phenomena in crystals. http://www4.hcmut.edu.vn/~huynhqlinh/project/Minhhoa3/Nhieuxa/Nx2/www.matter.org.uk/diffraction/geometry/2D_reciprocal_lattices.htm Chan Park, MSE-SNU. Intro to Crystallography, 2021. https://www.xtal.iqfr.csic.es/Cristalografia/index-en.html. 5. What is the reciprocal lattice?  When we need to consider certain planes in a crystal structure, it is more convenient to use surface normals rather than two-dimensional crystal planes.  In the stereographic projection, a pole (which represent a set of crystal planes) can be described by a point.  In the stereographic projection, the relative position of the planes and interplanar angles can be determined from the relative position of the poles. But the interplanar spacing (d), which is needed to determine the position of the diffracted X-ray (the θ in Bragg’s law), cannot be obtained from the relative position of the poles.. Real space. Reciprocal space.  Reciprocal lattice includes information on both the relative position of the planes and the interplanar spacing.. (10 0) (01 0). ⃗ Points  atomic positions Chan Park, MSE-SNU. Intro to Crystallography, 2021. ∗ ∗. Points  (hkl), reflections. 6. What is the reciprocal lattice?  Families of planes in crystals can be represented simply by their normal, which are then specified as (reciprocal lattice (RL)) vectors.  RL vectors can be used to define a pattern of (RL) points, each (RL) point representing a family of planes.  Advantage ; RL accentuates the connections between families of planes in the crystals, Bragg's law and the directions of the diffracted or reflected beams.. 1 d. 1 d. (010) O. ⃗ d010 Chan Park, MSE-SNU. d010. (010) (020). d010 7. Intro to Crystallography, 2021. Reciprocal lattice 1/d1. Reciprocal lattice point 1/d2. Chan Park, MSE-SNU. Intro to Crystallography, 2021. direction  orientation of (hkl) magntiude  interplanar spacing. Hammond Chap 6. 8. 5 plane lattices and their reciprocal lattices. (1). (010). (3). (100). hexagonal oblique. (2). (010). (100). rectangular. (5) (4) centered rectangular Chan Park, MSE-SNU. Intro to Crystallography, 2021. Chan Park, MSE-SNU. Intro to Crystallography, 2021. square Hammond Chap 6. 9. 10. Real vs. Reciprocal lattice. Chan Park, MSE-SNU. 11. Sherwood & Cooper Chapter 8.7. Intro to Crystallography, 2021. Reciprocal lattice → →. →. →. ∗. →. ∗. →. →. ⃗. ∗. →. ⃗. ⃗. ℎ ⃗. ℎ. ℎ. →. ℎ ⃗ ⃗. ⃗. ℎ. ℎ ∗. →. ℎ ⃗. (hkl). →. ⃗. ⃗. ℎ. ⃗. ∗. ⃗. ℎ. ⃗. ∗. ⃗. ∗. ∗. ∗. ⃗. ⃗. ∗. 1. ∗. 0. ∗. 0. ∗. 0. ∗. 1. ∗. 0. ∗. 0. ∗. 0. ∗. 1. Sherwood & Cooper Chapter 8.7 Chan Park, MSE-SNU. Intro to Crystallography, 2021. Hammond Chap 6. 12. a, b, c vs. a*, b*, c* ⃗. ∗. ⃗. ∗. ⃗. ∗. ⃗. ∗. 1. ∗. 0. ∗. 0. ∗. 0. ∗. 1. ∗. 0. ∗. 0. ∗. 0. ∗. 1. Chan Park, MSE-SNU. Intro to Crystallography, 2021. Hammond Chap 6. 13. Krawitz page 64. 14. Reciprocal lattice b1, b2, b3; reciprocal lattice. Chan Park, MSE-SNU. Intro to Crystallography, 2021. Reciprocal lattice. Chan Park, MSE-SNU. 15. Intro to Crystallography, 2021. Reciprocal lattice of a Monoclinic P. a ≠ b ≠ c α = γ = 90 o ≠ β. Fig 6.4. Chan Park, MSE-SNU. Intro to Crystallography, 2021. Hammond Chap 6. 16. Reciprocal lattice of a Monoclinic P. Reciprocal lattice unit cell of a monoclinic P crystal. Chan Park, MSE-SNU. Hammond Chap 6. Intro to Crystallography, 2021. 17. Direct lattice vector vs Reciprocal lattice vector  Direction symbols [uvw] are the components of a vector ruvw in direct space (direct lattice vector). ruvw = u a + v b + w c  Laue indices* are simply the components of a reciprocal lattice vector. d*hkl = h a* + k b* + l c*. *Laue indices, see page 141 of Hammond Chan Park, MSE-SNU. Intro to Crystallography, 2021. 18. Reciprocal lattice of a Cubic - I Fig 6.7. d200 = a/2  1/d200 = 2/a d110 = √2a/2  1/d110 = 2/√2a = √2/a d220 = √2a/4  1/d220 = 4/√2a = 2√2/a. Cubic I direct lattice - Cubic F reciprocal lattice Cubic F direct lattice - Cubic I reciprocal lattice Chan Park, MSE-SNU. Hammond Chap 6. Intro to Crystallography, 2021. 19. Real vs Reciprocal lattice in Cubic. Real space. Reciprocal space. Cubic F (rhombohedral 60°). Cubic I (rhombohedral 109.47°). Cubic P (rhombohedral 90°). Cubic P (rhombohedral 90°). Cubic I (rhombehodral 109.47°). Cubic F (rhombohedral 60°). Real lattice. Fourier transform. Reciprocal lattice. https://journals.iucr.org/j/issues/2014/05/00/nb5114/nb5114fig1.html Chan Park, MSE-SNU. Intro to Crystallography, 2021. See Hammond Fig. 3.2. 20. Reciprocal lattice. Real space. Reciprocal space. (100). ∗ ∗. ⃗. (010). Points  atomic positions. Chan Park, MSE-SNU. Points  (hkl), reflections. 21. Intro to Crystallography, 2021. Reciprocal lattice  dhkl is the vector drawn from the origin of the unit cell to intersect the first crystallographic plane in the family (hkl) at a 90° angle  The reciprocal vector d*hkl  |d*hkl| = 1/dhkl  In the reciprocal lattice, each point represents a vector which, in turn, represents a set of Bragg planes  Each reciprocal vector can be resolved into the components:. b*030. 130. 230. 330. 020. 120. 220. 320. 010. 110. 210. 310. 100. 200. 300. 400. 2 10. 3 10. 4 10. d*hkl= ha* + kb* + lc*. 0 10 1 10. a*. 020 Chan Park, MSE-SNU. Intro to Crystallography, 2021. Scott A Speakman. 22. Chan Park, MSE-SNU. Sherwood 1st ed. page 276. Intro to Crystallography, 2021. 23. Primitive lattices of Cubic F & Cubic I. bcc vs rhombohedral. fcc vs rhombohedral. cI.  a, b, c; conventional unit cell vectors  a’, b’, c’; corresponding primitive unit cell vectors. bR. c a’. b’ s. a. ⃗. ⃗. aR bI. aI. cR. b. c’. ⃗, ⃗, ; orthogonal unit vectors s; length of side of the cube. 1 1 1 a R = − a I + bI + c I 2 2 2 1 1 1 bR = a I − bI + cI 2 2 2 1 1 1 cR = a I + bI − cI 2 2 2. aI = 0aR + 1bR + 1cR bI = 1aR + 0bR + 1cR c I = 1aR + 1bR + 0cR. Sherwood 1st ed. Fig 8.9 Chan Park, MSE-SNU. Intro to Crystallography, 2021. Read Sherwood & Cooper 8.7. 24. When real lattice is non-primitive (Cubic-F). c a’. b’ s. b a. c’ c a*. b* a i, j, k; orthogonal unit vectors s; length of the side of the cube. c*. 2/s b. Reciprocal lattice of fcc is bcc. Sherwood 1st ed page 279 ~282 Chan Park, MSE-SNU. Read Sherwood Chap 8. Intro to Crystallography, 2021. 25. Reciprocal lattice vector r*hkl is normal to (hkl). A Chan Park, MSE-SNU. r*hkl & C. Intro to Crystallography, 2021. r*hkl  r*hkl. (hkl) Krawits page 65. 26. Chan Park, MSE-SNU. 27. Krawits page 66. Intro to Crystallography, 2021. Reciprocal lattice and interplanar spacing dhkl origin. → ℎ. * &'(). = ℎ%. ∗#. =. "∗. ∗ #! ∗. "∗. =. $ "∗. ∗. $ !. !. dhkl. (hkl). %. "∗. = (ℎ. ∗%. ∗. % ∗%. ∗. ∗). % ∗%. 2. (ℎ. ∗. ∗ ∗ cos . ∗. ∗. ∗). 2. ∗ ∗ cos / ∗. 2. ∗ ∗ cos 0 ∗. For cubic, a=b=c, α=β=γ=90o, a*=b*=c*=1/a, α*=β*=γ*=90o, a*•a* = 1/a2. %. $ !. * &'(). Chan Park, MSE-SNU. =. * # * #! * *. Intro to Crystallography, 2021. !. ℎ%. %. %. Cubic 28. Interplanar spacing. dhkl. Chan Park, MSE-SNU. Intro to Crystallography, 2021. 29. Interplanar spacing. dhkl. Chan Park, MSE-SNU. Intro to Crystallography, 2021. 30. Zonal equation * The plane (khl) which cuts the origin has the equation:. hx + ky + lz = 0 * A point uvw on the plane passing through the origin:. hu + kv + lw = 0. Zonal equation. * Two lattice lines [u1v1w1] and [u2v2w2] lie in the lattice plane (hkl) whose indices can be determined from the zonal equation:. hu1 + kv1 + lw1 = 0. hu2 + kv2 + lw2 = 0. [u2v2w2] [u1v1w1]. Chan Park, MSE-SNU. (hkl). ex) [101] and [121] (111) plane 31. Intro to Crystallography, 2021. Zonal equation, Weiss zone law. hu+kv+lw=n. hu+kv+lw=1 hu+kv+lw=0. Chan Park, MSE-SNU. Intro to Crystallography, 2021. Hammond Chap 6. 32. Zonal equation, Weiss zone law  (hkl) lies in a zone [uvw]  d*hkl ⊥ ruvw  d*hkl • ruvw = 0 ( h a* + k b* + l c*) • (u a + v b + w c) = 0  hu + ku + lw = 0. ( a * • a = 1, a * • b = 0).  For the lattice plane passing thru the origin, hu + ku + lw = 0  For the nth lattice plane, ruvw•i = ndhkl (i = d*hkl / |d*hkl | = d*hkl dhkl ). i. ruvw• d*hkl dhkl = ndhkl  ruvw• d*hkl = n hu + ku + lw = n. Chan Park, MSE-SNU. Intro to Crystallography, 2021. Hammond Chap 6. 33. Interplanar angles. Chan Park, MSE-SNU. Intro to Crystallography, 2021. 34. Interplanar angles. Chan Park, MSE-SNU. Intro to Crystallography, 2021. 35. Reciprocal lattice a, b, c: real lattice constants, a*, b*, c*; reciprocal lattice constants. V* = 1/V Chan Park, MSE-SNU. Intro to Crystallography, 2021. 36. Reciprocal lattices of Cubic and Hexagonal systems. Cullity page 40 Chan Park, MSE-SNU. Intro to Crystallography, 2021. 37. todos  Must Read All  Hammond - Chapter 6, A5  Krawitz - Chapter 2.8, 2.9, 2.10  Sherwood & Cooper - Chapter 8.7 (page 269 ~ 274; < 6 pages)  Cullity - Chapter 2-4, A1-1, A1-2, A1-3  Ott – Chapter 13.3.  Reciprocal lattice HW (due in 1 week)  Hammond page 162 – draw the reciprocal lattice sections of z = ½ and z = 1 from direct lattices of cubic-I and cubic-F  Ott 13.1  Cullity 2-10, 3-4  Hammond 5.1; 5.2; 5.3; 5.5; 6.1 (2nd part only; cubic-I and cubic-F)  What is Laue index? What is the difference between Laue Index and Miller Index? (See Hammond chap 5.5) Chan Park, MSE-SNU. Intro to Crystallography, 2021. 38.