1. D. Sherwood, Crystals, X-rays, and

Chapter 12 Diffraction

Reading Assignment:

1. D. Sherwood, Crystals, X-rays, and

Proteins-chapter 8

Contents

Non-normal Incident Waves Non-normal Incident Waves

Diffraction Pattern by 3-D Lattice Diffraction Pattern by 3-D Lattice

Laue Equation Laue Equation 3

1 2

4

Diffraction Pattern of a Crystal Diffraction Pattern of a Crystal

5 Reciprocal LatticeReciprocal Lattice

6 Ewald Circle/SphereEwald Circle/Sphere

Diffraction by 3-D Lattice

- diffraction pattern of a crystal amplitude function of a crystal

( ) ( ) *[ ( ) ( )]

fourier transform

( ) ( ) [ ( ) * (

f crystal f motif f infinite lattice f shape function Tf crystal Tf motif Tf infinite lattice Tf sha

=

=

i i

2

2 2

)]

diffraction pattern intensity

( )

( ) ( ) * ( )

- diffraction pattern of an infinite, 1-D array of -function in real space an i

pe function

Tf crystal

Tf motif Tf infinite lattice Tf shape function

δ

=

=

→ i

nfinite, 1-D array of -function in Fourier, or reciprocal space

δ

- infinite crystal- infinite 3-D array of -function

diffraction pattern- infinite 3-D array of -function - real lattice vs. reciprocal lattice

- real crystal- finite infinitely sharp peak blurred b

δ

δ

→ y shape function

- intensity of main peak- motif- information of structure of unit cell

- 1D- scattering angle sin

3D- wave vector - spatial coordinatesk

θ

Diffraction by 3-D Lattice

all

- non-normal incident waves ( )= ( )

normal incidence- : phase difference

- path difference sin - sin

2 ( sin - sin ) phase difference=

( )

-

d i

i k r r

F k f r d r

k r

OC BA r r

r r

k k r k r

e

β α

π β α

λ

= − =

= − = Δ

∫

i

i i

all

( )= ( ) ^{i k r}

r

F Δk

∫

e

Diffraction by 3-D Lattice

D. Sherwood, Crystals, X-rays, and Proteins

- non-normal incident waves

: scattering vector : scattering angle

d i

k k k

θ

Δ = −

Diffraction by 3-D Lattice

D. Sherwood, Crystals, X-rays, and Proteins

, ,

all

- diffraction pattern of finite 3-D lattice ( )= ( )

- for a lattice, unit vectors , , lattice point

amplitude function ( ) ( [

all p q r

i k r r

F k f r d r

a b c r pa qb rc

f r r p

e

δ

= + +

= −

∫

∑

i

, ,

) , ,

(

])

- ( ) ( [ ])

all p q r

all p q r

all p all q all r

i k r all r

i k p a qb r c

i k p a i k qb i k r c

a qb rc

F k r pa qb rc

e

e

e e e

δ

Δ + +

Δ Δ Δ

+ +

Δ = − + +

=

=

∫ ∑

∑

∑ ∑ ∑

i

i

i i i

Diffraction by 3-D Lattice

2 2 2 2

2 2 2

- ( )

sin sin sin

2 2 2

- ( )

sin sin sin

2 2 2

all p all q all r

i k p a i k qb i k r c

F k

P k a Q k a R k a F k

k a k a k a

e

e

e

Δ =

Δ Δ Δ

Δ =

Δ Δ Δ

∑

∑

∑

i i i

i i i

Diffraction by 3-D Lattice

D. Sherwood, Crystals, X-rays, and Proteins

2 2 2 2

2 2 2

sin sin sin

2 2 2

- ( )

sin sin sin

2 2 2

- first term, maximum at 0, , 2 , 2

2 ( : integer)

- first term, zero at

2 peak width

P k a Q k a R k a F k

k a k a k a

k a

k a h h

P k a

π π π

π

Δ Δ Δ

Δ =

Δ Δ Δ

Δ = ⋅⋅⋅

→ Δ =

Δ = ±

→ Δ

i i i

i i i

i i

i

( ) 4

- , peak becomes narrower function k a P

P

π

δ

Δ =

→ ∞ →

i

Diffraction by 3-D Lattice

2 2

2

2 2

2

2

- infinite crystal

sin 2

( 2 )

sin 2

- ( ) ( 2 ) ( 2 )

( 2 ) - real lattice vs.

all h

all h all k

all l

P k a

k a h k a

F k k a h k b k

k c l

δ π

δ π δ π

δ π

Δ → ⎡⎢ Δ − ⎤⎥

Δ ⎣ ⎦

⎡ ⎤ ⎡ ⎤

Δ = ⎢⎣ Δ − ⎥ ⎢⎦ ⎣ Δ − ⎥⎦

⎡ ⎤

× ⎢⎣ Δ − ⎥⎦

∑

∑ ∑

∑

i i i

i i

i

reciprocal lattice

Diffraction by 3-D Lattice

0

0 0

0

0

- ( 2 ) ( ) : unit vector

- ( ) refers to the point located at - what does ( ) represent?

finite only when projection of along i

all h all n

k a h r x nx x

r r r r

r x x

r x x r x

δ π δ

δ

δ

Δ − → −

− =

−

= →

∑

∑

i

i

0

0

plane

a stack of planes, all parallel to

s cons plane, separated by

tant equal to

yz x

x

⇒

Diffraction by 3-D Lattice

D. Sherwood, Crystals, X-rays, and Proteins

0 0

0 0

0 0 0

s

- ( ) plane, ( ) plane

- ( ) ( ) line

- ( ) ( ) ( ) point

all n all m

all n all m

all n all m all

r x nx r y my

r x nx r y my

r x nx r y my r z sz

δ δ

δ δ

δ δ δ

− → − →

− − →

− − − →

∑ ∑

∑ ∑

∑ ∑ ∑

i i

i i

i i i

Diffraction by 3-D Lattice

D. Sherwood, Crystals, X-rays, and Proteins

2 2 2

- ( 2 ) : a set of planes in space, perpendicular to the direction defined by a and separated by a distance 2 /a

- ( ) ( 2 ) ( 2 )

all h

all h all k

k a h k

F k k a h k b k

δ π

π

δ π δ π

Δ − Δ

⎡ ⎤ ⎡ ⎤

Δ = ⎢⎣ Δ − ⎥ ⎢⎦ ⎣ Δ − ⎥⎦

∑

∑ ∑

i

i i

2

( 2 )

three sets of planes define space lattice reciprocal lattice

all l

k c l

δ π

⎡ ⎤

×⎢⎣ Δ − ⎥⎦

→

∑

Diffraction by 3-D Lattice

Laue Equations

- 2 , 2 , 2

- what we seek is that value or those values, of the scattering vector which satisfy all three equations simultaneously.

-

- real space

d i

k a h k b k k c l

k

k k k

r pa qb

π π π

Δ = Δ = Δ =

Δ

Δ = −

= +

i i i

Fourier space (reciprocal space) (h * k * l *)

- (h * k * l *) 2

h * k * l * 2

h * k * l * 2

h * k * l * 2

rc

k a b c

k a a b c a h

a a b a c a h

a b b b c b k

a c b c c c l

χ

χ π

χ χ χ π

χ χ χ π

χ χ χ π

+

Δ = + +

Δ = + + =

+ + =

+ + =

+ + =

i i

i i i

i i i

i i i

Laue Equations

- =2

- h * k * l *

h * k * l * h * k * l *

- * 1 * =0 * 0 * perpendicular to &

* 0 * 1 * 0 *

a b b b c b k a c b c c c l

a a b a c a a b c

a b b b c b a b

χ π

ξ

+ + =

+ + =

+ + =

= = ⇒

= = = = ×

i i i

i i i

i i i

i i i

i i i

* 0 * 0 * 1 * =1 - h= , k= , l= *

- k 2 ( * * *)

c

a c b c c c a a a b c

h k l a b c

a b c ha kb lc

ξ

π

= = = = ×

= ×

×

Δ = + +

i i i i i

i

- * , * , *

- k 2 ( * * *)

G * * *

k 2 G

b c c a a b

a b c

a b c a b c a b c ha kb lc

ha kb lc π

π Δ =

× × ×

= = =

× × ×

Δ = + +

= + +

i i i

Laue Equations

Reciprocal Lattice

- real lattice is primitive direction

length angle

D. Sherwood, Crystals, X-rays, and Proteins

http://www.matter.org.uk/diffraction/geometry/2D_reciprocal_lattices.htm

- monoclinic P a ≠ ≠b c

α γ

β

Reciprocal Lattice

C. Hammond, The Basics of Crystallography and Diffraction

- primitive orthorhombic lattice

3 ₁

3

2 5 ¹

5 1

2

3 5 2 30 V = × × =

Reciprocal Lattice

D. Sherwood, Crystals, X-rays, and Proteins

- real lattice- non-primitive

if we use the conventional crystallographic base vectors, the reciprocal lattice so generated is not complete

as we have not taken into account those lattice sites at fractional distance within the unit cell

- choose three vectors which are compatible with given real lattice but which define a primitive unit cell

Reciprocal Lattice

aR

bR

cR

aI b^I

cI

1 ( )

2

1 ( )

2

1 ( )

2

a a x y z

b a x y z

c a x y z

= − + +

= − +

= + −

*

*

*

2 ( )

2 ( )

2 ( )

a y z

a

b z x

a

c x y

a π π π

= +

= +

= +

a*

b*

c*

Reciprocal Lattice

C. Kittel, Introduction to Solid State Physics

- cubic I a = =b c

90^o

α β γ

Reciprocal Lattice

C. Hammond, The Basics of Crystallography and Diffraction

Reciprocal Lattice Direction vs. Real Lattice Plane

- Theorem 1

The reciprocal lattice vector

* * *

is perpendicular to the ( ) set of planes in real lattice.

- proof ( 1 1, , 0) * * *

Ghkl ha kb lc

hkl

AB G a b c

h k ^αβγ

α β γ

= + +

= − = + +

1 1

( ) ( * * *)

=0 if perpendicular

AB G a b a b c

h k h k

αβγ = − +

α

β

γ

α β

i i

AC

h k

h l

α β

α γ

∴ =

⇒ =

(hkl) plane closest to origin

D. Sherwood, Crystals, X-rays, and Proteins

Reciprocal Lattice Direction vs. Real Lattice Plane

- is perpendicular to ( ) plane if - one solution , ,

- * * * is normal to ( ) set of planes - another solution

, , * *

hkl

nhnknl

G hkl

h k l

h k l

G ha kb lc hkl

nh nk nl G nha nkb n

αβγ

α β γ

α β γ

α β γ

= =

= = =

= + +

= = = ⇒ = + + *

lc

D. Sherwood, Crystals, X-rays, and Proteins

Reciprocal Lattice Direction vs. Real Lattice Plane - Theorem 2

The magnitude of the reciprocal vector is related to the spacing between the ( ) set of planes by

1

- proof

* * *

hkl hkl hkl

hkl

hkl

hkl hkl

hkl

G G

d hkl

G d

a G a ha kb lc

d h G h

=

+ +

= i = i

1

hkl

hkl

G

G

=

D. Sherwood, Crystals, X-rays, and Proteins

http://www.matter.org.uk/diffraction/geometry

Bragg’s Law

-

2 sin , 2 2

2 sin

- Bragg's

la 2

2 2

2 si

w

n _hk

d i

hkl

hkl

l hkl

k k k

k k k

k G

d

d

k d θ k λ

θ π

λ π π

π π

θ

Δ = −

Δ = =

Δ = =

⇒ = ⇒ = =

D. Sherwood, Crystals, X-rays, and Proteins

We normally set n=1 and adjust Miller indices, to give 2d_hkl sin θ = λ

2d sin θ = nλ

e.g. X-rays with wavelength 1.54Å are reflected from planes with d=1.2Å. Calculate the Bragg angle, θ, for constructive interference.

λ = 1.54 x 10^-10 m, d = 1.2 x 10^-10 m, θ=?

⎟ ⎠

⎜ ⎞

⎝

= ⎛ λ θ

λ

= θ

−

d 2 sin n

n sin

d 2

1

n=1 : θ = 39.9°

n=2 : X (nλ/2d)>1

Example of equivalence of the two forms of Bragg’s law:

Calculate θ for λ=1.54 Å, cubic crystal, a=5Å 2d sin θ = nλ

(1 0 0) reflection, d=5 Å n=1, θ=8.86^o

n=2, θ=17.93^o n=3, θ=27.52^o n=4, θ=38.02^o n=5, θ=50.35^o n=6, θ=67.52^o

no reflection for n≥7

(2 0 0) reflection, d=2.5 Å

n=1, θ=17.93^o

n=2, θ=38.02^o

n=3, θ=67.52^o

no reflection for n≥4

Ewald Circle

- Bragg's law 2 sin

- consider a two dimensional system crystal- two dimensional real lattice two dimensional reciprocal lattice - , , confined to a plane

- 2 - 1

- CO

hkl

i d

i d

d

k k k

k k

r

θ λ

π λ λ

=

⇒

Δ

= =

=

: incident wave, CP : diffracted wave OP : scattering vector

D. Sherwood, Crystals, X-rays, and Proteins

Ewald Circle

- 2 sin 2 1 sin

2 - 2

2 1

- sin 2 sin

- Ewald circle or Reflecting circle

hl

hk hk

hk

k OP r

k G

G d

d

θ θ

λ π

π

θ θ λ

λ

= = = Δ

Δ =

= = ⇒ =

D. Sherwood, Crystals, X-rays, and Proteins

Reciprocal Lattice and Diffraction

- assume the orientation of the inciddent beam w.r.t. the real lattice, and hence w.r.t. the reciprocal lattice

- O: origin of the reciprocal lattice

- superimpose the reciprocal lattice on the Ewald circle

- reciprocal lattice vector Ghk

- Ewald circle, Bragg condition

D. Sherwood, Crystals, X-rays, and Proteins

- rotating crystal

Reciprocal Lattice and Diffraction

D. Sherwood, Crystals, X-rays, and Proteins

- limiting circle 2 2

2 2

sin , 0 sin 1, 0

hk

hk hk

G r

G G

λ

θ θ

λ λ

≤ =

= ≤ ≤ ≤ ≤

Reciprocal Lattice and Diffraction

D. Sherwood, Crystals, X-rays, and Proteins

Why X-Ray Diffraction Works

o

-1

o

o

- 2

- crystal- length of unit cell: 1 nm (10 A)

magnitude of reciprocal lattice vector: 1 (nm) - 2 , 2 nm

- greatest wavelength- 2 nm (20 A)

- typical X-ray wavelength- 1 A satisfy the

hk

hk

G

G

λ

λ λ

≤

≤ ≤

inequality

Ewald Sphere

- 2D 3

Ewald circle Ewald sphere

- before Ewald sphere construction

1. direction of incident beam relative to real crystal 2. geometric property of reciprocal lattice

- rules for Ewald sphere constr

→ D

→

uction

i) draw a sphere of radius ( 1/ ) about (crystal)

ii) draw a diameter (incident beam direction) w.r.t. crystal iii) choose the origin of reciprocal lattice as point

iv) plot t

r C

ICO

O

λ

=

he reciprocal lattice

Ewald Sphere

- ensure that the origin of the reciprocal lattice is chosen as that point at which the incident beam leaves the Ewald sphere

ensure that the scale used for the Ewald sphere is same as that used for the reciprocal lattice

D. Sherwood, Crystals, X-rays, and Proteins

Ewald Sphere and Diffraction

D. Sherwood, Crystals, X-rays, and Proteins

Ewald Sphere and Diffraction

- limiting sphere

D. Sherwood, Crystals, X-rays, and Proteins

Rotation Camera

- crystal is mounted with crystallographic axis accurately vertical

crystal rotates about the axis, and diffraction pattern is recorded on the cylinderical film

- ex) c-axis in orthorhombic crystal

Laue Cone

D. Sherwood, Crystals, X-rays, and Proteins

Rotation Camera

- symmetrical in both position and intensity about equator and also meridian

D. Sherwood, Crystals, X-rays, and Proteins

Powder Camera

- speimen-fine powder- a vrey large number of randomly oriented crystallite

- reciprocal sphere

D. Sherwood, Crystals, X-rays, and Proteins

Bragg’s Law and Crystal Planes

- Bragg's law 2 sin

- geometric relationship between , , - 2

( ) set of planes ( ) set of planes line AX is parallel to the (hkl) set of plane

hkl

i d

hkl hkl

hkl

d

k k k

k G k G

G hkl k hkl

θ λ

π

=

Δ

Δ = ⇒ Δ

⊥ ⇒ Δ ⊥

∴ s

reflection

D. Sherwood, Crystals, X-rays, and Proteins

Bragg’s Law and Crystal Planes

hkl

- a physical explanation of Bragg's law - path difference=2d sin

- phase difference= 2 2 sin

4 sin

- maximum intensity 2 2 sin

hkl

hkl

hkl

d

d n d n

θ

π θ

λ

λ

π θ π

λ

D. Sherwood, Crystals, X-rays, and Proteins

Effect of Finite Crystal Size

- finite size of the crystal- blurring or smearing out diffraction pattern → rather fuzzy, not a point

streaked or blurred image

weak diffraction pattern

D. Sherwood, Crystals, X-rays, and Proteins