1

Reciprocal Space

The reciprocal lattice is based on the Bravais lattice (not crystal lattice) in direct space.

The collection of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice (eiKir = e iKi(R+r) ) is known as its reciprocal lattice.

Beginning Chapter 5

K belongs to the reciprocal lattice of a Bravais lattice of points R, provided that the relation eiKiR = 1 holds for all R.

Let a1, a2, and a3 be a set of primitive vectors for the direct lattice. The reciprocal lattice can be generated by the three primitive vectors

as can be verified by

)(2;

)(2;

)(2

321

213

321

132

321

321 aaa

aabaaa

aab

aaaaa

b rrr

rrrrrr

rrrrrr

rrr

×⋅×

=×⋅

×=

×⋅×

= πππ

ba πδ2rry

The reciprocal lattice is itself a Bravais lattice.

Any vector in real or reciprocal space can be expressed in terms of the respective primitive vectors:

332211 anananR rrrr++=

332211 bkbkbkkrrrr

++=)(2 332211 nknknkRk ++=⋅ π

rr

ijji ba πδ2=⋅

Reciprocal of the Reciprocal Lattice

The reciprocal lattice of any reciprocal lattice (K) is the original direct Bravais lattice (R), since all vectors R for which the relation eiKiR = 1 holds for all K belong to the original direct lattice and all points of the direct lattice satisfy thebelong to the original direct lattice, and all points of the direct lattice satisfy the relation eiKiR = 1 for all K.

Specifically, using vector identity , it can be shown that

and so on.

The volume of a unit cell in the reciprocal lattice is

)(2

321

321 bbb

bba rrr

rrr

×⋅

×= π

)()()( BACCABCBArrrrrrrrr

⋅−⋅=××

|)(|4

)]([)()(4

321

12

2321

21132

32 aaaa

aaaaaaa

bb rrr

r

rrr

rrrrrr

×⋅=

×⋅×××

=×ππ

The volume of a unit cell in the reciprocal lattice is

The larger the unit cell in direct space, the smaller is the unit cell in reciprocal space.

cellcell Vaaa

bbb3

321

3

321

8|)(|

8|)(|

ππ=

×⋅=×⋅=Ω rrr

rrr

2

Examples

Simple Cubic zaayaaxaa ˆ;ˆ;ˆ 321 ===rrr z

aby

abx

ab ˆ2;ˆ2;ˆ2

321πππ

===rrr

Face-Centered Cubic

Body-Centered Cubic

)ˆˆ(2

;)ˆˆ(2

;)ˆˆ(2 321 yxaaxzaazyaa +=+=+=

rrr

)ˆˆˆ(2;)ˆˆˆ(2;)ˆˆˆ(2321 zyx

abzyx

abzyx

ab −+=+−=++−=

πππ rrr

)ˆˆˆ(2

;)ˆˆˆ(2

;)ˆˆˆ(2 321 zyxaazyxaazyxaa −+=+−=++−=

rrr

)ˆˆ(2;)ˆˆ(2;)ˆˆ(2321 yx

abxz

abzy

ab +=+=+=

πππ rrr

Other Examples

Monoclinic

90o rotation

angle and ratio conserved

Hexagonal: rules similar to monoclinic’s

Orthorhombic: bi = 2π/ai

Centered Structures: In general, reciprocal cell dimension doubles in all directions affected by the additional points. Reciprocal lattice also becomes centered.

Body centered Face-centered

Base-Centered Base-centered

3

Brillouin Zones

The Wigner-Seitz primitive cell of the reciprocal lattice is the first Brillouin Zone.

The nth Brilloun zone is the region in reciprocal space reachable from the origin (Γ point) by crossing over (n-1) Bragg planes that is unreachable by crossing over only (n-2) Bragg planes.

A i fi i i i i

6th

The “origin” of the reciprocal lattice is known as the Γ-point.

Alternative definition: For a point k in reciprocal space, draw spheres of radius |k| about every reciprocal lattice point. If k is in the interior of n-1 spheres and on the surface of one, then it lies in the interior of the nth Brillouin zone.

3rdWhat if a point lies on the surface of more than one spheres?

Brillouin Zones

4

Lattice Planes

Vectors in reciprocal lattice are related to lattice planes in direct space.

A lattice plane is any plane containing at least three noncollinear Bravais lattice points. Any such plane actually contains an infinite number of points.

Family of lattice planes: set of parallel, equally spaced lattice planes, which together contain all the points of the three-dimensional Bravais lattice.

For any family of lattice planes separated by a distance d, there are reciprocal lattice vectors perpendicular to the planes, the shortest of which have a length of 2π/d. Conversely, for any reciprocal lattice vector K, there is a family of lattice planes normal to K and separated by a distance d, where 2π/d is the length of the shortest reciprocal lattice vector parallel to K.

Miller Indices of Lattice Planes

The Miller indices (h, k, l) of a set of lattice planes, with an inter-planar spacing of d, are the coordinates of the reciprocal lattice vector normal to that plane, with a magnitude equaling 2π/d, i.e. k=hb1+ kb2+ lb3. This is a reciprocal space

fi i idefinition.

The Miller indices also have a definition in real space. With one lattice plane assumed to cut through the origin, the intersections of the adjacent plane on the three axes of the direct axes are used to define (h, k, l) as (note that xi may be infinity) :

332211 ,, axaxax rrr

1,1,1 lkh ===

0 used for infinite interceptis used for negative h, k, l

321

,,xxx

lkh ,,

5

Cubic Lattice Planes

common cubic planes

Distance between adjacent crystal planes:

CUBIC222 lkh

adhkl++

=

Non-Cubic Systems Miller Indices

2-D Lattice (Monoclinic)

orthorhombic2

2

2

2

2

2

2

1cl

bk

ah

dhkl

++=

(Monoclinic)

23

2

212

222

2

221

2

2 sincos2

sinsin1

al

aahk

ak

ah

dhkl

+−+=γγ

γγ

tetragonal 2

2

2

22

2

1cl

akh

dhkl

++

=

hkl

Areal Density of Lattice Points on Plane = n * dhkl

6

Hexagonal Systems

4 indices

(h h h l)(h1 h2 h3 l)

sum of first three indices always vanishes

X-ray Diffraction

1. X-ray well suited for determination of crystal structures. 1 angstrom wavelength 12.3 keV

hc

Beginning Chapter 6

2. Bragg found discrete intense peaks (Bragg peaks) of scattered radiation from crystalline materials for certain energy and incident directions.

3. Bragg accounted for sharp scattered peaks as radiation reflected off of ordered crystallographic planes specularly and constructively.

λω hc=h

Bragg Conditionnλ = 2d sinθ

7

Why does a plane reflect specularly?

Suppose a plane wave with wave vector k impinges upon a collection of atoms on a single plane. Assuming that atoms scatter x-ray elastically (k=k’) and spherically symmetric, what is the amplitude of wave at a point r far away from the sample along the direction k’ ?

⋅′ rki rr)( Rrki j

rrr−⋅′

independent of in-plane

arrangement of atoms

∑ ⋅′−

j

Rkkirki

jer

e rrr)(

||)(

)(

j

Rrki

j

Rki

RreerA

jj rr

r rr

−∝ ⋅∑

large r

jRkkrrr

⊥′− )(Constructive interference if

von Laue’s Formulation of XRD

Von Laue did not assume specular reflection from crystal planes, but general scattering from identical units.g

First consider just two scattering centers displaced by d. The optical path difference is

which, for constructive interference, givesk, k’, d are not coplanar, in general

)ˆˆ(coscos nnddd ′−⋅=′+r

θθ

...,2,1,)ˆˆ( ±±==′−⋅ mmnnd λr mkkd π2)( =′−⋅

rrr

λπ2|||| =′= kk

rr

For the entire crystal to contribute constructively, the displacement between any two scattering centers also satisfies

for all R,

which simply states that the vector K = k-k’ belongs to the reciprocal lattice.mkkR π2)( =′−⋅

rrr 1)( =′−⋅ kkRierrr

8

Bragg Plane

A Bragg plane is any plane that is the perpendicular bisector of the line joining the origin of k-space (Γ-point) to a reciprocal lattice point. Von Laue’s condition

ifi i iff i i

π/2-θ

specifies constructive diffraction whenever the change in wave vector is a reciprocal vector, K= k-k’. Since |k|=|k’|, the three vectors k, k’, and K form a triangle with two sides equal. If the k vector is drawn from the Γ-point, its tip falls on the plane bisecting Γ and K.

Equivalence of the Bragg and von Laue conditions

||ˆ21 KKk

rr=⋅

Note: K does not need to be a “primitive” reciprocal lattice vector. What happens when K is not a primitive reciprocal vector?

q gg

Bragg Conditionnλ = 2d sinθ

von Laue Condition nK = 2k sinθ

In real space, what is the orientation of the planes that reflect the X-rays?

multiply both sides by 2π/(kK)

Ewald Construction

EWALD CONSTRUCTION

Given the incident wave vector k, a sphere of

Laue Method of crystal structure determination.G ve e c de w ve vec o , sp e e o

radius k is drawn about the point k. Any reciprocal lattice vector falling on the surface of this Ewald sphere leads to Bragg reflection. Note that this is a condition which is not easily met for a randomly oriented single crystal with a monochromatic x-ray beam.

A “white” x-ray source (wide range of

wavelengths or energies)

9

X-Ray Diffraction (XRD)

Rotating Crystal Method

Monochromatic X rays, with crystal rotated.

i fi i

Debye-Scherrer Method (Powder)Monochromatic X rays, with powder sample.

The Ewald sphere is fixed in space.

φ21sin2|| kK =

r

Theta - 2 Theta Method

Detector scans at twice the angular speed of the sample.

Theta - 2 theta scan is used to identify phases of the sample (powder diffraction pattern). It is also used to measure the lattice spacing perpendicular to the surface.

10

X-Ray Rocking Curve

Also known as double-crystal diffraction, x-ray rocking curve is

d t th “ iused to measure the “mosaic spread” of microcrystals or the overall quality of a crystal (thin film).

X-Ray Diffraction Structure Factor

Geometrical Structure Factor: lattice with a basis (of identical atoms).

Intensity of Bragg peak proportional

Example: b.c.c. considered as s.c. with basis

jdKin

jK eS

rr⋅

=∑=

1

Intensity of Bragg peak proportional to |SK|2 in additional to other angular dependencies.

( )nynxnK ˆˆˆ2++

πr

[ ])ˆˆˆ(exp1 21 zyxaKiSK ++⋅+=

r

( )znynxna

K 321 ++=

⎭⎬⎫

⎩⎨⎧

++++

=−+= ++

oddnnnevennnn

S nnnK

321

321

,0,2

)1(1 321

same as f.c.c. lattice with double the cell linear dimension

11

Face Centered Cubic

X-Ray Diffraction Structure Factor

312321 )1()1()1(1 nnnnnnKS +++ −+−+−+=

⎬⎫

⎨⎧

=oddallorevenallnnn

S 321 ,,,4

Monatomic Diamond Lattice

[ ])ˆˆˆ(exp1 41 zyxaKiSK ++⋅+=

r

( )znynxna

K ˆˆˆ2321 ++=

πr

⎪⎫

⎪⎧ ++ 4,2 321 bydivisibleisnnn

⎭⎬

⎩⎨=

oddandevenmixednnnSK

321 ,,,0not applicable to (2,1,1)

⎪⎭

⎪⎬

⎪⎩

⎪⎨

×=++++±=

2,0,1

321

321

numberoddnnnoddisnnniSK

Hexagonal Close-Packed, Centered Orthorhombic, Centered Tetragonal, etc.

not applicable to (2,2,1)

Atomic Form Factor

jdKin

eKfSrrr ⋅∑ )( j

jjK eKfS

=∑=

1)(

∫ ⋅−= )(1)( rerd

eKf j

rKij

rrr rr

ρ

Variation in the intensity of Bragg peaks can help distinguish crystal structures.structures.

12

Sample Shape Reciprocal Point Shape

Suppose a plane wave with wave vector k impinges upon a crystal of finite size.

332211,, 321anananR nnnrrvr

++= 333222111 ;; NnNNnNNnN ≤≤−≤≤−≤≤−

∑ ⋅′−⋅′

321

3,2,1

,,

)(

nnn

Rkkirki

nnner

e rrrrr

||)(

321

3,2,1

321

3,2,1

,,

)(

,, nnn

Rrki

nnn

Rki

RreerA

nnnnnn rr

rrrr

rr

−∝

−⋅′⋅∑

large r

Assuming that atoms scatter x-ray elastically and spherically symmetric,

If the sample is thin, ∞== 321 ;10~ NNN

Express 332211)( bxbxbxkkrrrrr

++=′−

∑∑∑∑∞

−∞=

∞

−∞=−=

⋅′− ∝3

33

2

221

11

11

321

3,2,1 )2()2()2(

,,

)(

n

ixn

n

ixnN

Nn

ixn

nnn

Rkki eeee nnn πππrrr

Delta functions

Sample Shape Reciprocal Point Shape

Reciprocal lattice consists of lattice “points” because the crystal is assumed to be infinite in size, i.e. reciprocal lattice is th F i t f f l i fi it l tti d ithe Fourier transform of a real infinite lattice, and vice versa.

If the sample under study has a finite size (microscopic size) along one or more of its dimension, the reciprocal lattice also takes on finite width along those directions.

Examples: thin film rods, thin wire discs, small particles blobs, etc. Ewald sphere drawn as usual, but chances of

cutting across some reciprocal lattice “points” are greatly g p p g yenhanced.

13

Surface Diffraction

a layer of regularly arranged atoms ……

….. is transformed into an array of rods in reciprocal space.

The array of rods in reciprocal space is then used in an Ewald construction of

the diffraction pattern.