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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. How to construct a reciprocal lattice from a crystal
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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
How to construct a reciprocal lattice from a crystal (1) Pick a set of planes in a crystal parallel Plane set 2 d2 d1 Plane set 1 Does it really form a lattice? Draw it to convince yourself! Plane set 3 d3
Example: a monoclinic crystal Reciprocal lattice (a* and c*) on the plane containing a and c vectors. (b is out of the plane) c b a (-100) (100) c (001) (102) c (001) (002) O (002) O a O a (00-2) c (00-2) (002) O a c (001) c* c (101) (002) * (002) O a* a (10-1) (00-1) a (00-2) 2D form a 3-D reciprocal lattice
Lattice point in reciprocal space Integer Lattice points in real space
Reciprocal lattice cells for cubic crystals: Simple cubic: z y x (002)? z z (010) (100) y y a a x x a (110) (001) Simple cubic y y x x
(111) (111) O O (111) FCC corner Up and down O F and B L and R (222)
You should do the same for a FCC and show it forms a BCC lattice! (Homework!) Vector: dot and cross product v v.u= |v||u|cos Projection of v onto u and times each other (scaler)! u |v|cos vu= |v||u|sin w |v|sin |v||u|sin is the area of the parallelogram. w v and u u
Relationships between a, b, c and a*, b*, c*: Monoclinic: plane y-axis (b) : cc*. c c* Similarly, d001 b a cc* = |c*|ccos, |c*| = 1/d001ccos = d001 cc* = 1
Similarly, aa* = 1 and bb* =1. c* //ab, Define c* = k (ab), k : a constant. cc* = 1 ck(ab) = 1 k= 1/[c(ab)]=1/V. V: volume of the unit cell Similarly, one gets 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 ruvw = 0
nth plane ruvw Define the unit vector in the d*hkl direction i, r2 d*hkl uvw r1
Reciprocal Lattice: Fourier transform of the spatial wavefunctionof the original lattice wave process (e. g. electromagnetic) in the crystal Crystal: periodic Physical properties function of a crystal Crystal translation vector Periodic function Exponential Fourier Series
u, v, w: integer Translation vectors of the original crystal lattice for all T h, k, l: integer If Vectors of the reciprocal lattice always integer If k(reciprocal) lattice ; T original lattice! Vice versa!
In crystallography In SSP k (in general): momentum space vector; G: reciprocal lattice points
Proof: the reciprocal lattice of BCC is FCC Use primitive translation vectors only BCC FCC z y x corner Up and down F and B L and R
The vector set is the same as the FCC primitive translation vector. Unit of the reciprocal lattice is 1/length.
Mathematics of Interference Sum of two waves: assume A1 = A2 = A R new amplitude AR
Geometrical analysis of Interference term rotation vector A
Complex Wave Representation of Interference assume A1 = A2 = A
Diffraction conditions and reciprocal lattices: Theorem: The set of reciprocal lattice vectors G determines the possible X-ray reflections. k r r k http://en.wikipedia.org/wiki/Plane_wave k
Complex exponential form Complex number What happen to the time dependent term? X-ray wavelength ~ 0.1 nm ~ 3x1018 1/s Detectors get the average intensity! Detectors measured the intensity only! A lot of time, examining is enough!
Path difference Similarly, Phase angle dV Phase angle = (2/)rsin = kr r O Fourier expansion n(r) 0 for G = k otherwise, = 0 The diffraction condition is G = k. k + G = k’
G: reciprocal lattice, -G: reciprocal lattice? ____ Bragg condition? k G (hkl) plane or k dhkl k Bragg law G= k.
More geometric relation between direct lattice and reciprocal lattice: e1, e2, e3: contravariant basis vector of R3 covariant basis vectors e1, e2, e3 (reciprocal lattice) eiand ei are not normal, but mutually orthonormal: For any vector v:
vcan 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!
Use the BCC lattice as examples Assume You can check the other way around.
where Similarly, Prove
gij: metric tensorin direct lattice a, b, c and , , : direct lattice parameters (standard definition of the Bravais lattice) det|gij| = V2.
Inverting the matrixgij. gij : metric tensor in reciprocal lattice a*, b*, c* and *, *, *: reciprocal lattice parameters These two are the same!
One gets relation like Similarly, ……………..
Example：FCC BCC • (1) Find the primitive unit cell of the selected • structure • (2) Identify the unit vectors
Volume of F.C.C. is a3. There are four atoms per unit cell! the volume for the primitive of a F.C.C. structure is ?
Similarly, B.C.C. See page 23
Using primitive translation vector to do the reciprocal lattice calculation: Case: FCC BCC
not Why? (hkl) defined using unit cell! (hkl) is defined using primitive cell! (HKL)
Example 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
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 2k b a b/k a/h h A/(h+2k) or
Interplanar spacing (defined based on unit cell) Cubic: Tetragonal: Orthorombic: Hexagonal:
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!
