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Expression of d-dpacing in lattice parameters. September 18, 2007.  d-spacing of lattice planes ( hkl ):. For cubic, a=b=c. Finally, one can get the d -spacing of ( hkl ) plane in any crystal.  The process has constructed the reciprocal lattice points

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slide4

 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

the following is additional

The following is additional

Reciprocal lattice

(not required in CENG 151 syllabus)

slide6

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

slide7

(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

slide8

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)

slide9

 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.

slide10

(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

slide11

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)

slide12

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)

slide13

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

slide14

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?

slide15

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.

slide16

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

slide17

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.

slide18

* 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

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