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Reciprocal lattice How to construct reciprocal lattice Meaning of reciprocal lattice Relation between reciprocal lattice and diffraction Geometrical relation between reciprocal lattice and original lattice

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How to construct a reciprocal lattice from a crystal (1) Pick a set of planes in a crystal Plane set 1 Plane set 2 d1d1 d2d2 parallel d3d3 Plane set 3 Does it really form a lattice? Draw it to convince yourself!

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O a c (001) (002) (00-2) (100) (-100) O O a c (001) (002) (00-2) (101) O a c (001) (002) (00-2) (102) O a c (002) (00-1) (10-1) ** a c a* c* Example: a monoclinic crystal Reciprocal lattice (a* and c*) on the plane containing a and c vectors. (b is out of the plane) a c b 2D form a 3-D reciprocal lattice

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Lattice point in reciprocal space Lattice points in real space Integer

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x y z Reciprocal lattice cells for cubic crystals: Simple cubic: x y z (100) x y z (010) x y (001) x y (110) a a a Simple cubic (002)?

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Base centered cubic (BCC): x y z

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(111) O O O FCC cornerUp and down F and BL and R (222)

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You should do the same for a FCC and show it forms a BCC lattice! (Homework!) Vector: dot and cross product u v v.u = |v||u|cos |v|cos Projection of v onto u and times each other (scaler)! v u = |v||u|sin w |v|sin u |v||u|sin is the area of the parallelogram. w v and u

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Relationships between a, b, c and a *, b *, c * : Monoclinic: plane y-axis (b) a c c*c* d 001 Similarly, c c * = |c * |ccos , |c * | = 1/d 001 ccos = d 001 c c * = 1 : c c *. b

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Similarly, a a * = 1 and b b * =1. c * //a b, Define c * = k (a b), k : a constant. c c * = 1 c k(a b) = 1 k = 1/[c (a b)]=1/V. Similarly, one gets V: volume of the unit cell The Weiss zone law or zone equation: A plane (hkl) lies in a zone [uvw] (the plane contains the direction [uvw]). d * hkl (hkl) d * hkl r uvw = 0

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uvw d * hkl r uvw r1r1 Define the unit vector in the d * hkl direction i, r2r2 nth plane

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Crystal translation vector Reciprocal Lattice: Fourier transform of the spatial wavefunction of the original lattice wave process (e. g. electromagnetic) in the crystal Physical properties function of a crystal Crystal: periodic Periodic function Exponential Fourier Series

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Translation vectors of the original crystal lattice for all T u, v, w: integer If Vectors of the reciprocal lattice h, k, l: integer always integer If k (reciprocal) lattice ; T original lattice! Vice versa!

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In crystallography In SSP k (in general): momentum space vector; G: reciprocal lattice points

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Proof: the reciprocal lattice of BCC is FCC Use primitive translation vectors only BCC x y z FCC cornerUp and down F and BL and R

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BCC

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The vector set is the same as the FCC primitive translation vector. Unit of the reciprocal lattice is 1/length.

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Mathematics of Interference Sum of two waves: assume A 1 = A 2 = A new amplitude A R RR

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Geometrical analysis of Interference term rotation vector A

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Complex Wave Representation of Interference assume A 1 = A 2 = A

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Diffraction conditions and reciprocal lattices: Theorem: The set of reciprocal lattice vectors G determines the possible X-ray reflections. r k r k k

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A lot of time, examining is enough! Complex exponential form Complex number What happen to the time dependent term? X-ray wavelength ~ 0.1 nm ~ 3x /s Detectors get the average intensity! Detectors measured the intensity only!

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O r Phase angle = (2 / )rsin = k r Similarly,Phase angle dV Fourier expansion n(r) The diffraction condition is G = k. k + G = k ’ 0 for G = k otherwise, = 0 Path difference

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G: reciprocal lattice, -G: reciprocal lattice? ____ Bragg condition? (hkl) plane G d hkl k k kk or Bragg law G = k.

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More geometric relation between direct lattice and reciprocal lattice: e 1, e 2, e 3 : contravariant basis vector of R 3 covariant basis vectors e 1, e 2, e 3 (reciprocal lattice) e i and e i are not normal, but mutually orthonormal: For any vector v:

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v can be expressed in two (reciprocal) ways: Einstein’s summation convention, omitting No proof here, but you can check whether these relation is correct or not? Use BCC or FCC lattice as examples, next page!

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Use the BCC lattice as examples Assume You can check the other way around.

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where Similarly, Prove

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g ij : metric tensor in direct lattice a, b, c and , , : direct lattice parameters (standard definition of the Bravais lattice) det|g ij | = V 2.

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Matrix inversion:

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Inverting the matrix g ij. g ij : metric tensor in reciprocal lattice a *, b *, c * and *, *, * : reciprocal lattice parameters These two are the same!

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One gets relation like Similarly, ……………..

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d-spacing of (hkl) plane for any crystal system

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Example ： FCC BCC (1) Find the primitive unit cell of the selected structure (2) Identify the unit vectors

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Volume of F.C.C. is a 3. There are four atoms per unit cell! the volume for the primitive of a F.C.C. structure is ?

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Similarly, B.C.C. See page 23

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Using primitive translation vector to do the reciprocal lattice calculation: Case: FCC BCC

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notWhy? (hkl) defined using unit cell! (hkl) is defined using primitive cell! (HKL)

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Find out the relation between the (hkl) and [uvw] in the unit cell defined by and the (HKL) and [UVW] in the unit cell defined by. In terms of matrix Example

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Find out the relation between (hkl) and (HKL). Assume there is the first plane intersecting the a axis at a/h and the b axis at b/k. In the length of |a|, there are h planes. In the length of |b|, there are k planes. How many planes can be inserted in the length |A|? Ans. h + 2k H = 1h + 2k + 0l Similarly, K = -1h + 1k +0l and L = 0h + 0k + 1l A B a b a/ha/h b/kb/k 2k2k h A/(h+2k) or

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There are the same! Or

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We proof the other way around!

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Interplanar spacing (defined based on unit cell) Cubic: Tetragonal: Orthorombic: Hexagonal:

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Get the metric tensor! Perform the inversion of the matrix! Comparing the inversion of the metric tensor in direct lattice with the metric tensor in reciprocal lattice Geometrical relation between reciprocal lattice and direct lattice can be obtained!

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