Expression of d-dpacing in lattice parameters

1. Expression of d-dpacing in lattice parameters September 18, 2007

2.  d-spacing of lattice planes (hkl): For cubic, a=b=c

3. Finally, one can get the d-spacing of (hkl) plane in any crystal

4.  The process has constructed the reciprocal lattice points (do form a lattice), which also shows the reciprocal lattice unit cell for this section outlined by a* and c*. One can extend this section to other sections , see To form a 3D reciprocal lattice with in reciprocal space in real space

5. The following is additional Reciprocal lattice (not required in CENG 151 syllabus)

6. Reciprocal lattice Introduction:  The reciprocal lattice vectors define a vector space that enables many useful geometric calculations to be performed in crystallography. Particularly useful in finding the relations for the interplanar angles, spacings, and cell volumes for the non-cubic systems.  Physical meaning: is the k-space to the real crystal (like frequency and time), is the real to Fourier variables.  Let’s start first with the less elegant approach. One has to have basic knowledge of vectors and their rules. Reciprocal lattice vectors:  Consider a family of planes in a crystal, the planes can be specified by two quantities: (1) orientation in the crystal

7. (2) their d-spacings. The direction of the plane is defined by their normals.  reciprocal lattice vector: with direction || plane normal and magnitude  1/(d-spacing). Plane set 2 d2 d1 Plane set 1 k: proportional constant, taken to be a value with physical meaning, such as in diffraction, wavelength  is usually assigned. 2dsin =   /d = 2sin. Longer vector  smaller spacing  larger . Plane set 3 Is it really form a lattice? Draw it to convince yourself! d3

8. Reciprocal lattice unit cells:  Use a monoclinic crystal as an example. Exam the reciprocal lattice vectors in a section perpendicular to the y-axis, i.e. reciprocal lattice (a* and c*) on the plane containing a and c vectors. (-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)

9.  The process has constructed the reciprocal lattice points (do form a lattice), which also shows the reciprocal lattice unit cell for this section outlined by a* and c*. One can extend this section to other sections , see To form a 3D reciprocal lattice with in reciprocal space in real space Reciprocal lattice cells for cubic crystals:  The reciprocal lattice unit cell of a simple cubic is a simple cubic. What is the reciprocal lattice of a non- primitive unit cell? For example, BCC and FCC? As an example. Look at the reciprocal lattice of a BCC crystal on x-y plane.

10. (010) O y In BCC crystal, the first plane encountered in the x-axis is (200) instead of (100). The same for y-axis. (200) (100) (110) 1/2 x (020) O  Get a reciprocal lattice with a centered atom on the surface. The same for each surface. Exam the center point. In BCC, the first plane encountered in the (111) direction is (222).  FCC unit cell

11. 022 222 002  Reciprocal lattice of BCC crystal is a FCC cell. No 111 011 101 220 110 000 200 In FCC the first plane encountered in the x-axis is (200) instead of (100). The same for y-axis. But, the first plane encountered in the diagonal direction is (220) instead of (110). Centered point disappear 1/2 y (220) (110) x In FCC, the first plane encountered in the [111] direction is (111). (111)

12. 022 222 002 202  Reciprocal lattice of FCC crystal is a BCC cell. 111 020 220 000 200  Another way to look at the reciprocal relation is the inverse axial angles (rhombohedral axes). FCC SC BCC Real 60o 90o 109.47o. Reciprocal 109.47o 90o 60o  In real space, one can defined the environments around lattice points In terms of Voronoi polyhedra (or Wigner -Seitz cells. The same definition for the environments around reciprocal lattice points  Brillouin zones. (useful in SSP)

13. Proofs of some geometrical relationships using reciprocal lattice vectors:  Relationships between a, b, c and a*, b*, c*: See Fig. 6.9. Plane of a monoclinic unit cell  to y-axis. : angle between c and c*. c  c* d001 a Similarly, Consider the scalar product cc* = c|c*|cos, since |c*| = 1/d001 by definition and ccos = d001  cc* = 1 Similarly, aa* = 1 and bb* =1. Since c* //ab, one can define a proportional constant k, so that c* = k (ab). Now, cc* = 1  ck(ab) = 1  k = 1/[c(ab)] 1/V. V: volume of the unit cell

14. Similarly, one gets  The addition rule: the addition of reciprocal lattice vectors  The Weiss zone law or zone equation: A plane (hkl) lies in a zone [uvw]  the plane contains the direction [uvw]. Since the reciprocal vectors d*hkl  the plane  d*hkl ruvw = 0  uvw lies on the plane through the origin When a lattice point uvw lies on the n-th plane from the origin, what is the relation?

15. ruvw Define the unit vector in the d*hkl direction i, d*hkl uvw r1  d-spacing of lattice planes (hkl):  The rest angle between plane normals, zone axis at intersection of planes, and a plane containing two directions. See text or part four.

16. Reciprocal lattice in Physics:  In order to describe physical processes in crystals more easily, in particular wave phenomena, the crystal lattice constructed with unit vectors in real space is associated with some periodic structure called the reciprocal lattice. Note that the reciprocal lattice vectors have dimensions of inverse length. The space where the reciprocal lattice exists is called reciprocal space. The question arises: what are the points that make a reciprocal space? Or in other words: what vector connects two arbitrary points of reciprocal space?  Consider a wave process associated with the propagation of some field (e. g., electromagnetic) to be observed in the crystal. Any spatial distribution of the field is, generally, represented by the superposition of plane

17. waves such as The concept of a reciprocal lattice is used because all physical properties of an ideal crystal are described by functions whose periodicity is the same as that of this lattice. If φ(r) is such a function (the charge density, the electric potential, etc.), then obviously, We expand the function φ(r) as a three dimensional Fourier series * This series of k (some uses G) defined the reciprocal lattice which corresponds to the real space lattice. R is the translational symmetry of the crystal.

18. * Thus, any function describing a physical property of an ideal crystal can be expanded as a Fourier series where the vector G runs over all points of the reciprocal lattice * What is the meaning of this equation? is the phase of a wave exp(ikR)=1  kR=2n, some defined the reciprocal lattice as