Expression of d-dpacing in lattice parameters

September 18, 2007

d-spacing of lattice planes (hkl):

))((1 ******2

** cbacbadd lkhlkhdhkl

hklhkl

2

2

2

2

2

2**2**2**2

2

1

c

l

b

k

a

hlkh

dhkl

ccbbaa

2 2 2

2 20

1

hkl

h k l

d a

For cubic, a=b=c

0

2 2 2hkl

ad

h k l

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

********

2*22*22*2********

cos2cos2*cos2

)()(

alhccklbbhka

clbkahlkhlkhhklhkl

cbacbadd

2

2

2

2

2

2

2

2222

2

2222

2

2222

2

)coscos(cos2

)coscos(cos2

)coscos(cos2

sinsinsin1

V

cablh

V

bcakl

V

abchk

V

bal

V

cak

V

cbh

dhkl

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

*001

**010

**100

* ;; ddbda c****

001*010

*100

* cbadddd lkhlkhhkl in reciprocal spacecbar wvuuvw in real space

The following is additional

Reciprocal lattice(not required in CENG 151 syllabus)

Reciprocal latticeIntroduction: 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

(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 1

Planeset 2

d1

d2

*1d

*2d

)/1(:)/1(: 21*2

*1 dddd

2

*2

1

*1 ;

d

k

d

k dd

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 .

d3

Planeset 3

*1d

*2d

*3d

Is it really form a lattice?Draw it to convinceyourself!

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.

Oa

c(001)

(002)

(00-2)

(100)

(-100)

O

*001d

*002d

*100d

*100d

Oa

c(001)

(002)

(00-2)

(101)

Oa

c(001)

(002)

(00-2)

(102) *101d

*102d

Oa

c

(002)

(002)

(00-1) (10-1)

*110d

*

a

c

a*

c*

*001

**100

* ; dda c

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

*001

**010

**100

* ;; ddbda c****

001*010

*100

* cbadddd lkhlkhhkl in reciprocal spacecbar wvuuvw 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.

O

x

y

1/2

(010)

(020)

(200)(100)

In BCC crystal, the first planeencountered in the x-axis is (200)instead of (100). The same fory-axis.

O*020d *

040d

*200d

*400d

*110d

Get a reciprocal lattice with a centered atom on the surface.The same for each surface.

(110)

Exam the center point. In BCC, the first plane encountered in the (111) direction is (222). FCC unit cell

000

002

200

022222

011

101

110 220

No 111 Reciprocal lattice of BCC crystal is a FCC cell.

x

y1/2 In FCC the first plane encountered

in the x-axis is (200) instead of(100). The same for y-axis. But, thefirst plane encountered in thediagonal direction is (220) insteadof (110). Centered point disappear

(110)(220)

In FCC, the first plane encountered in the [111] direction is (111).

(111)

000

002

200

022222

220

Reciprocal lattice of FCC crystal is a BCC cell. 111

202

020

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)

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*.

a

cc*

d0010 and 0 and **** bcacbcac

bac //*

Similarly, cbacaba // ;0 and 0 ***

acbcbab // ;0 and 0 ***

Consider the scalar product cc* = c|c*|cos,since |c*| = 1/d001 by definition and ccos = d001

cc* = 1Similarly, aa* = 1 and bb* =1.Since c* //ab, one can define a proportional constant k,so that c* = k (ab). Now, cc* = 1 ck(ab) = 1 k = 1/[c(ab)] 1/V. V: volume of the unit cell

V

bac

* Similarly, one gets

VV

acb

cba

** ;

The addition rule: the addition of reciprocal lattice vectors

*))()((

*)(

*)( 212121222111 nlmlnkmknhmhlkhnlkhm ddd

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

0

0)()( ***

lwkvhu

wvulkh cbacba

When a lattice point uvw lies on the n-th plane from theorigin, what is the relation?

uvw lies on the plane through the origin

uvwd*

hkl

ruvw

r1

2*

21** )( rdrrdrd hklhkluvwhkl

Define the unit vector in the d*hkl

direction i, 1 ;

||*

*

*

iidd

di hklhkl

hkl

hkl d

nnddnd hklhklhklhklhkl *2

*22 drdrir

nlwkvhunuvwhkl rd *

d-spacing of lattice planes (hkl):

))((1 ******2

** cbacbadd lkhlkhdhkl

hklhkl

2

2

2

2

2

2**2**2**2

2

1

c

l

b

k

a

hlkh

dhkl

ccbbaa

The rest angle between plane normals, zone axis at intersection of planes, and a plane containing two directions. See text or part four.

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

waves such as rk ik e

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 thislattice. If φ(r) is such a function (the charge density, the electric potential, etc.), then obviously,

)()( Rrr We expand the function φ(r) as a three dimensionalFourier series

k

Rkrkk

k

rkk Rrr iii eee )()( 1 Rkie

* 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.

* What is the meaning of this equation? is the phase of a wave exp(ikR)=1 kR=2n, some defined the reciprocal lattice as

)exp( Rk i

* 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

G

rGGr ie )(

VVV

bac

acb

cba

2 ;2 ;2 ***