Understanding Reciprocal Lattice in Crystallography
Delve into the concept of reciprocal lattice in crystallography, exploring how the orientation of planes is defined, the relationship between direct and reciprocal space, the significance of Bragg's law in diffraction patterns, the practical use of reciprocal space in experimental measurements, and the process of indexing diffraction patterns to determine the reciprocal unit cell and ultimately the direct unit cell.
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Presentation Transcript
RECIPROCAL LATTICE The orientation of a plane is defined by the direction of a normal to the plane: = d N d hkl hkl 2
RECIPROCAL LATTICE dhkl vectors in the direct space = d N d [ ] hkl hkl d*hkl vectors in the reciprocal space N = * d [1/ ] hkl d hkl 3
RECIPROCAL LATTICE The reciprocal lattice is periodic. Therefore, we should be able to define the reciprocal unit cell: (a*, b*, c*) such that any vector in the lattice can be described as d*hkl = ha* + kb* + lc* Now let s see how the reciprocal lattice is related to the diffraction pattern 4
Why Use The Reciprocal Space? Think about the Bragg law: = 2 hklsin n d What do we measure experimentally? 1 1 n n n n n = = = = = * hkl sin sin d 2 2 2 2 2 d d d d hkl hkl hkl hkl Angles at which diffraction maxima are observed 5
Why Use The Reciprocal Space? r2 r1 x 1 n = = * sin d x hkl 2 r 2 The experimental measurement is directly related to d*hkl OR The detector is scanning for the reciprocal lattice points d*hkl 6
Why Use The Reciprocal Space? A diffraction pattern is not a direct representation of the crystal lattice The diffraction pattern is a representation of the reciprocal lattice In order to find the reciprocal lattice, the diffraction pattern can be indexed Diffraction from a single Xtal 7
INDEXING PROCEDURE To index a diffraction pattern means: to find such a basis (a*,b*,c*) that all the diffraction spots (or lines) can be described (indexed) as d*hkl = ha* + kb* + lc* with only integer (hkl) values allowed. All we need to do is to find the value of d* for each spot and then let software find the unit cell. Basically, we deal with a system of linear equations. The more reflections we have, the more reliably the unit cell will be determined. 8
THE RECIPROCAL UNIT CELL The reciprocal UC is related to the direct UC: By definition which means that a* (bc) (ab) a* a* a= 1 b* a = 0 c* a = 0 a* b= 0 b* b = 1 c* b = 0 a* c= 0 b* c = 0 c* c = 1 b b* (ac) c* c If we can index the diffraction pattern and find the reciprocal UC, then we will be able to find the direct UC. 9
THE RECIPROCAL UNIT CELL Draw the reciprocal unit cell (b axis is to the plane of the paper) 10
diffraction crystal planes - (100), (200), (100) (200) (300) Families of planes Lattice plane directions- (100) (100) (200)
3. Reciprocal lattice c a b c a b = * = = b * * c a V + V V h *= / 1 r d = + * * * * r a k b c l hkl 102 102 112 112 A reciprocal lattice point corresponds to a diffraction (lattice) plane of its original lattice. A reciprocal vector r* is perpendicular to a lattice plane with the indices (hkl). 101 101 111 111 100 100 110 110 a* a* 001 001 c* c* b* b* 000 000 010 010 r* r* 1 1 22 22 1 1 01 01 1 1 11 11 1 1 10 10 1 1 00 00
