Transcript Expression of d-dpacing in lattice parameters

Expression of d-dpacing in
lattice parameters
September 18, 2007
 d-spacing of lattice planes (hkl):
d
*
hkl
d
*
hkl
1
 2  (ha *  kb*  lc * )( ha *  kb*  lc * )
d hkl
1
h2 k 2 l 2
2 *
*
2 *
*
2 *
*
 h a a  k b b  l c c  2  2  2
2
d hkl
a b c
For cubic, a=b=c
d hkl 
1
h2  k 2  l 2

2
d hkl
a02
a0
h k l
2
2
2
Finally, one can get the d-spacing of (hkl) plane
in any crystal
d
*
hkl
d
*
hkl
2 *2
2 *2
2 *2
 (ha  kb  lc )  (ha  kb  lc )  h a  k b  l c
*
*
*
*
*
*
 2hka*b* cos  * 2klb*c* cos  *  2lhc *a* cos  *
2 2
2
2 2
2
2 2
2
1
2 b c sin 
2 a c sin 
2 a b sin 
h
k
l
2
2
2
d hkl
V
V
V2
abc 2 (cos  cos   cos  )
a 2bc(cos  cos   cos  )
 2hk
 2kl
2
V
V2
ab 2 c(cos  cos   cos  )
 2lh
V2
 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
*
*
*
a *  d100
; b*  d 010
; c*  d 001
*
*
*
*
d hkl
 hd100
 kd 010
 ld 001
 ha *  kb*  lc * in reciprocal space
ruvw  ua  vb  wc in real space
The following is additional
Reciprocal lattice
(not required in CENG 151 syllabus)
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
(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*
d1
d 2*
d1* : d 2*  (1 / d1 ) : (1 / d 2 )
k
k
*
*
d1  ; d 2 
d1
d2
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
d3
d 2*
d1* Is it really form a lattice?
Draw it to convince
d 3* yourself!
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
(102)
(001)
c
(001)
(002)

(002)
a
O
a
(00-2)
c
(00-2)
(002)
(001) (101) c
(002)

O
a
(00-2)
O
a
*
d 002
*
102
*
101
*
100
d
d
*
d 001
d
O
*
d10
1
O
c
(002)
(00-1)
*
d 00
1
(10-1)

a*
c*
*
a*
*
*
a  d100
; c*  d 001
 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
*
*
*
a *  d100
; b*  d 010
; c*  d 001
*
*
*
*
d hkl
 hd100
 kd 010
 ld 001
 ha *  kb*  lc * in reciprocal space
ruvw  ua  vb  wc 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 nonprimitive unit cell? For example, BCC and FCC?
As an example. Look at the reciprocal lattice
of a BCC crystal on x-y plane.
O
(010)
y
(200)
(100)
(110)
1/2
x (020)
O
d
*
200
*
*
d 020
d 040
*
d110
In BCC crystal, the first plane
encountered in the x-axis is (200)
instead of (100). The same for
y-axis.
 Get a reciprocal lattice with a
centered atom on the surface.
The same for each surface.
*
d 400
Exam the center point.
In BCC, the first plane
encountered in the (111) direction
is (222).  FCC unit cell
022
222
002
 Reciprocal lattice of BCC
crystal is a FCC cell.
No 111
011
101
000
220
110
200
1/2
(220)
(110)
x
y
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
In FCC, the first plane encountered
in the [111] direction is (111).
(111)
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)
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.
c
: angle between c and c*.
c*
 c*  a and c *  b  c*  a  0 and c*  b  0
c * // a  b
Similarly, a**  b  0 and a**  c  0; a** // b  c
b  a  0 and b  c  0; b // c  a
d001
a
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
ab
bc
ca
*
*
c 
; b 
Similarly, one gets a 
V
V
V
*
 The addition rule: the addition of reciprocal lattice
vectors
d m* ( h1k1l1 )  d n*( h2k2l2 )  d(*mh1 nh2 )( mk1 nk2 )( ml1 nl2 )
 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 
(ha *  kb*  lc * )  (ua  vb  wc )  0
 hu  kv  lw  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?
*
*
*
d hkl
 ruvw  d hkl
 (r1  r2 )  d hkl
 r2
Define the unit vector in the
direction i,
d*
hkl
ruvw
d*hkl
*
hkl
*
hkl
d
*
i
 d hkl
d hkl ; i  i  1
|d |
*
*
r2  i  nd hkl  r2  d hkl
d hkl  nd hkl  r2  d hkl
n
uvw
r1
*
 d hkl
 ruvw  n  hu  kv  lw  n
 d-spacing of lattice planes (hkl):
1
*
*
*
*
*
*

(
h
a

k
b

l
c
)(
h
a

k
b

l
c
)
2
d hkl
2
2
2
1
h
k
l
2 *
*
2 *
*
2 *
*

h
a

a

k
b

b

l
c

c
 2 2 2
2
d hkl
a b c
*
*
d hkl
 d hkl

 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  k  eikr
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,
 (r )   (r  R)
We expand the function φ(r) as a three dimensional
Fourier series
 (r )  k eik r   (r  R)  k eik r eik R
k
 eik R  1
k
* 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.
* 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
 (r )   G eiGr
G
* What is the meaning of this equation?
exp( ik  R) is the phase of a wave exp(ikR)=1 
kR=2n, some defined the reciprocal lattice as
bc
ca *
ab
*
a  2
; b  2
; c  2
V
V
V
*