1. 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

2. parallel d2 d1 Does it really form a lattice? Draw it to convince
 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

3. 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)
(001) c
(102)
(001) c
(002) O
(002)  a O O a
(00-2) c
(00-2)
(002) O a
(001) c c*
(101) c
(002) 
(002) * O  a a*
(10-1)
(00-1) a
(00-2)
2D form a 3-D reciprocal lattice

4. Lattice point in reciprocal space
Integer
Lattice points in real space

5. 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) y y
Simple cubic x x

6. Base centered cubic (BCC):z y x

7. (111)
(111) O O
(111)
FCC
corner
Up and down O
(222)
F and B
L and R

8. Vector: dot and cross product v v.u = |v||u|cos 
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
vu = |v||u|sin w
|v|sin 
|v||u|sin is the area of the
parallelogram. w  v and u u

9. Relationships between a, b, c and a*, b*, c*:
Monoclinic: plane  y-axis (b)
: c  c*. c  c*
Similarly, b
d001 a
cc* = |c*|ccos, |c*| = 1/d001  ccos = d001
 cc* = 1

10.  The Weiss zone law or zone equation:
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.
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 

11. Define the unit vector in the d*hkl direction i, d*hkl
nth plane
ruvw
Define the unit vector in the d*hkl
direction i,
d*hkl r2
uvw r1

12.  Reciprocal Lattice: Fourier transform of the spatial
wavefunction of 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

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

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

15. Proof: the reciprocal lattice of BCC is FCC
Use primitive translation vectors only
BCC
FCC x y z
corner
Up and down
F and B
L and R

16. BCC

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

18. Mathematics of Interference
Sum of two waves:
assume A1 = A2 = A R
new amplitude AR

19. Geometrical analysis of Interference
term  rotation vector A

20. Complex Wave Representation of Interference
assume A1 = A2 = A

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

22. 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!

23. Fourier expansion n(r)  0 for G = k otherwise, = 0
Path difference
Similarly,
Phase angle dV
Phase angle
= (2/)rsin
= kr r O 
Fourier expansion n(r)
 0 for G = k
otherwise, = 0
The diffraction condition is G = k.  k + G = k’

24. G: reciprocal lattice, -G: reciprocal lattice? ____
Bragg condition? k G
(hkl)
plane  or k k
dhkl
Bragg law
G = k.

25. 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)
ei and ei are not normal, but mutually orthonormal:
For any vector v:

26. 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!

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

28. where
Similarly,
Prove

29. gij: metric tensor in direct lattice
a, b, c and , ,  : direct lattice parameters
(standard definition of the Bravais lattice)
det|gij| = V2.

30. Matrix inversion:

31. Inverting the matrix gij.
gij : metric tensor in reciprocal lattice
a*, b*, c* and *, *, *: reciprocal lattice parameters
These two are the same!

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

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

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

35. 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 ?

36. Similarly,
 B.C.C.
See page 23

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

39. not
Why?
(hkl) defined using unit cell!
(hkl) is defined using primitive cell!
(HKL)

40.  Find out the relation between the (hkl) and
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

41. 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 b 2k a
b/k
a/h h
A/(h+2k) or

43. There are the same! Or

44. We proof the other way around!

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

46. 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!