Analytical Methods for Materials
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1 Analytical Methods for Materials Lesson 15 Reciprocal Lattices and Their Roles in Diffraction Studies Suggested Reading Chs. 2 and 6 in Tilley, Crystals and Crystal Structures, Wiley (2006) Ch. 6 M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007). Appendix A from Cullity & Stock 386
2 Corresponding to any crystal lattice in real space is a reciprocal lattice 387
3 Our interest stems from the fact that many properties are reciprocal to those of the crystal lattice. Do you remember what I said on Day 1 of the class: crystals diffraction is inverse to lattice spacings! 388
4 Fun in reciprocal space, The New Yorker Collection, John O Brien, from All rights reserved. Things can look very different in reciprocal space than in real space. 389
5 Things can look very different in reciprocal space than in real space. 390
6 Reciprocal Lattice Provides a vector representation of crystal directions and spacing between diffracting planes. Real Space: (lattice parameters define lattice) a, b, c,,, Reciprocal Space: (just another type of lattice) a, b, c,,, 391
7 FCC Crystal 2,0,2 0,0,2 c 2,2,2 0,2,2 1,1,1 a b 2,0,0 0,0,0 2,2,0 0,2,0 Real Space Reciprocal Space
8 Recall some simple vector operations Dot product (scalar product): AB A Bcos C h k l h k l Cross product (vector product): AB C A B sin A B C is the direction A B plane Other useful relations (and there are more ) AB C ( B A) B C B C C B A A A 393
9 Reciprocal Vectors c ab c a b c area of base unit cell volume 1 height 1 c d 001 d 001 in real space z O y x a c A b B C 394
10 Thus, we can define entire reciprocal lattices For an arbitrary lattice [a b c and( 90 )] in real space. a b c c b c c c bc a b V a c a b ca V ab ab ab V V unit cell volume a bc b ca c ab a bc plane in real space; it is the reciprocal lattice vector for a b ac plane in real space; it is the reciprocal lattice vector for b c ab plane in real space; it is the reciprocal lattice vecto r for c O c a c a b b 395
11 Example of the relationship between the real lattice and the reciprocal lattice. From Leng, p
12 Example of the relationship between the real lattice and the reciprocal lattice. From Leng, p
13 Useful Properties of the reciprocal lattice a b c 1 cos cos cos a cos d sin sin cos cos cos b cos d sin sin cos cos cos c cos d sinsin
14 Relationship between real and reciprocal In orthogonal real lattices (a= b = c and = = (= 90 )) a a; b b; c c Not necessarily so in non-orthogonal lattices We can draw an analogy between reciprocal and real lattices: r uavbwc uvw We use this to build a real lattice from unit cells r ha kb lc hkl We can use this principle to build a reciprocal lattice 399
15 Construction of a 2D reciprocal lattice a b (a) draw the plane lattice and mark the unit cell Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ,
16 Construction of a 2D reciprocal lattice Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, b a (b) draw lines perpendicular to the two sides of the unit cell to give the axial directions of the reciprocal lattice basis vectors. 401
17 Construction of a 2D reciprocal lattice Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, b a Sets axes and interaxial angle What you re doing first is finding the spacing between the planes making up the unit cell sides. (c) determine the perpendicular distances from the origin of the direct lattice to the end faces of the unit cell, d 10 and d 01, and take the inverse of these distances, 1/d 10 and 1/d 01, as the reciprocal lattice axial lengths, a and b. 402
18 Construction of a 2D reciprocal lattice Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, d a 01 b 1/ d Draw reciprocal lattice using axes a = 1/d 10 b = 1/d 01 Take reciprocals to get reciprocal lattice parameters. (d) mark the lattice points at the appropriate reciprocal distances, and complete the lattice. 403
19 Construction of a 2D reciprocal lattice Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, 2006 a 00 b 01 Every point on a reciprocal lattice represents a set of planes in the real space crystal! 00 1 a d b 1/ d d Real lattice Reciprocal lattice The vector joining the origin of the reciprocal lattice to a lattice point hk is perpendicular to the (hk) planes in the real lattice and of length 1/d hk. 404
20 c (a) a c c a d 001 d 100 (b) a Construction of a 3D reciprocal lattice 001 c c 1 d (c) a 1 d 100 a 001 (d) 002 c a Figure 2.10 The construction of a reciprocal lattice: (a) the a-c section of the unit cell in a monoclinic (mp) direct lattice; (b) reciprocal lattice aces lie perpendicular to the end faces of the direct cell; (c) reciprocal lattice points are spaced a = 1/d 100 and c = 1/d 001 ;(d) the lattice plane is completed by extending the lattice; (e) the reciprocal lattice is completed by adding layers above and below the first plane. 000 b 1 d Adapted from R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, Just like 2-D (e)
21 Cubic Reciprocal Lattice b 040 b 0.25 Å -1 (010) 4 Å (110) c 4 Å (210) REAL LATTICE ruvw uavbwc a [110] [010] [210] c RECIPROCAL LATTICE r ha kb lc hkl a Every point on a reciprocal lattice represents a set of planes in the real space crystal! Reciprocal lattice vectors are 90 away from realspace planes! 406
22 Hexagonal Reciprocal Lattice 000 b 0.25 Å -1 4 Å a 010 c (210) b 100 r 110 r (110) (010) 110 a (100) NORMAL LATTICE Every point on a reciprocal lattice represents a set of planes in the real space crystal! 210 RECIPROCAL LATTICE
23 What does the BCC reciprocal lattice look like? 2,0,1 0,0,2 1,1,2 2,2,2 0,2,2 1,0,1 2,1,1 c 0,1,1 b 1,2,1 2,0,0 0,0,0 1,1,0 0,2,0 2,2,0 Real Space Reciprocal Space
24 Importance of Reciprocal Space When a diffraction event occurs, the diffracted waves/pattern will match the reciprocal lattice. REASON: EM radiation scatters in inverse proportion to the spacing between diffraction centers (i.e., planes in crystals). 409
25 Bragg s Law: n 2dsin sin 2d sin hkl hkl Why use reciprocal space We can combine n and d as follows: d dhkl n This allows us to write Bragg's Law as: hkl 1 /2 dhkl d 2 hkl (Defines conditions where a crystal is oriented for coherent scattering) opposite hypotenuse 410
26 sin hkl 1 /2 dhkl opposite d 2 hkl hypotenuse BC AC Ewald Sphere Construction Reflection or Ewald sphere B 1 d hkl d hkl 2 Limiting sphere CRITICAL Only lattice points lying within the limiting sphere can diffract. Points lying on the Ewald sphere will satisfy the Bragg condition. A θ hkl (0,0,0) 2 2θ hkl 1 C Origin of reciprocal lattice and center of limiting sphere See this web site for an explanation and example 411
27 OB OB sin CO 2 / Which can be re-written as: 1 2 sin OB Since B is a reciprocal lattice point: Ewald s sphere C s o Crystalline solid A 1 B d 130 d 030 d 020 d 010 (0,0,0) O d 010 d 130 d 120 d 110 d 100 d 230 d 220 d 210 d OB dhkl g d hkl 1 dhkl and 2dhkl sin OB [Bragg s Law] RECIPROCAL SPACE Size of sphere corresponds to wavelength of radiation used (see next page). Rotation of the crystal will cause points to lie on the sphere. When points lie on the sphere, Bragg s law is satisfied! 412
28 If you change λ, you change the radius of the Ewald s sphere. This is how the Laue technique works. Diffractometers generally use fixed λ and variable θ. 201 reflected beam reflected beam 201 reflected beam / min 1/ max Reflecting sphere for smallest wavelength Reflecting sphere for largest wavelength Adapted from C. Hammond, The Basics of Crystallography and Diffraction, 3 rd Edition (Oxford University Press, Oxford, UK, 2009) p
29 Adapted from A.D. Krawitz, Introduction to Diffraction in Materials Science and Engineering, Wiley, New York, θ θ 2 Increasing diffraction angle Diffractometers generally use fixed λ and variable θ. 414
30 Synopsis 1. The reciprocal lattice allows us to compute the spacing between successive lattice planes in a crystal lattice. 2. The reciprocal lattice vector d hkl with components (hkl) is perpendicular to the plane with Miller indices (hkl). 3. The length of the reciprocal lattice vector is equal to the inverse of the spacing between the corresponding planes. 4. Diffraction of X-rays (and electrons) is described by the Bragg equation, which relates the radiation wavelength (λ) to the diffraction angle (θ) and the spacing between crystal planes (d hkl ). 415
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