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

(1)

Chapter 11 Diffraction

Reading Assignment:

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

Proteins-chapter 5 and 7

(2)

Diffraction by 1-D Obstacles

all

- geometric arrangement

diffraction pattern amplitude ( ) ( )= ( )

( ) : amplitude function

i k r r

F k

F k f r d r

f r

∫ e

ⁱ

plane

obstacle- along -axis r ( , 0, 0

scattering )

( ) ( )

( , 0, )

r sin : angle

x z

x

xz

x x

f r f x

k k k

k k x kx

θ θ

=

→

=

= =

i

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

(4)

sin -

2

all

- geometric arrangement

( )= ( ) = ( )

(sin ) ( ) ( : constant)

- in general, (sin ) is complex

intensity of diffraction pattern (sin )

ikx

i k r r

F k f r d r f x e dx

F f x e dx k

F

e

^θ

θ θ

θ

∞

→ =

∫ ∫

∫

i

∵

Diffraction by 1-D Obstacles

(5)

- one narrow slit

an infinite opaque sheet along the -axis containing narrow slit at the origin

- narrowness- compared to the wavelength - amplitude function- function

x

δ

Diffraction by 1-D Obstacles

(6)

sin sin

- -

sin

0 2

- ( ) ( )

- (sin ) ( ) = ( )

1

- intensity ~ (sin ) 1

- intensity is uniform at all angles

- a single narrow slit an active point source

ikx ikx

ikx

x

f x x

F f x e dx x e dx

e F

θ θ

θ

δ

θ δ

θ

∞ ∞

=

⎡ ⎤

= ⎣ ⎦ =

=

≡

∫ ∫

Diffraction by 1-D Obstacles

(7)

0 0

sin -

sin sin

0 0

- -

sin sin

- two narrow slit

a pair of functions, one at , and the other at

- ( ) ( ) ( )

- (sin ) ( )

( ) ( )

2

ikx

ikx ikx

x x

f x x x x x

F f x e dx

x x e dx x x e dx

e e

θ

θ θ

δ

δ δ

θ

δ δ

∞

∞ ∞

−

+ −

= + + −

=

= + + −

= + =

∫

∫ ∫

0 0

-

0

2 2

0

( ( ) ( ) ( )

cos( sin )

- (sin ) 4 cos (

)

sin ) kx

F

f x x x x x

d f x k

θ

θ θ

∞ δ

∞

− =

=

∫

∵

Diffraction by 1-D Obstacles

(8)

Interference

(9)

Young’s Experiment

maxima sin

d θ = mλ

(10)

Diffraction by 1-D Obstacles

0 0

sin -

0

2 2

0

- three narrow slits

( ) ( ) ( ) ( )

- (sin ) ( ) 1 2 cos( sin )

- (sin ) [1 2 cos( sin )]

ikx

f x x x x x x

F f x e dx

kx

F kx

θ

δ δ δ

θ

θ θ

∞

= + + + −

=

= +

∫

nine times more intense

(11)

0

- narrow slits

equally spaced by a distance function 2 1 (odd number)

- ( ) ( )

n p

n p

N

x N

N p

f x x nx

δ

= δ

=−

→

= +

=

∑

−

Diffraction by 1-D Obstacles

(12)

0 0 0

0 0

0

sin -

sin ( 1) sin sin

sin sin 2 sin

(2 1) sin s

sin sin

sin

- (sin ) ( )

1

(1 )

1 1

( ) (

1

ikx

ikpx ik p x ikpx

ikpx ikx ik px

ik p x ikNx

ikpx ikpx

ikx

F f x e dx

e e e

e e

e

θ

θ θ θ

θ

θ ^∞

∞

− − −

−

− + −

=

= + + ⋅⋅⋅ + + ⋅⋅⋅ +

= + + ⋅⋅⋅ +

− −

= =

−

∫

0

2 0

in sin

0

2

sin sin

2 sin sin

2

1 ) sin sin

2

sin sin

2 - (sin )

ikx

Nkx kx

e Nkx

kx

F

θ θ

θ

−

=

0 0

*main peak

angular width- 4 / separated by- 2 /

- 2 subsidiary peaks Nkx kx N

π π

Diffraction by 1-D Obstacles

(13)

0

- infinite number of narrow slits

separated by a distance infinite array of function

- ( ) ( )

- (sin ) (sin 2 )

n

x

f x x nx

F n

kx

δ δ

θ δ θ π

=∞

=−∞

=∞

=−∞

→

= −

∑

0

*infinitely sharp peak (sin ) 2

kx

θ π

Δ =

Diffraction by 1-D Obstacles

(14)

Grating Spectroscope

visible emission line of cadmium

visible emission line from hydrogen

sin

d θ = m λ

D. Halliday, Fundamentals of Physics

(15)

0

0 0

0

sin -

- one wide slit

an opaque screen containing a slit which is wide ~100

- ( ) 0 - -

1 - 0

- F(sin )= ( )

( )

ikx

X

ikx X

f x if x X

if X x X

if X x

f x e dx

θ

λ

θ ^∞

∞

= ∞ < <

< < + + < < ∞

=

∫

⁰

0

sin

sin - ikx X

X

e ik

θ

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

Diffraction by 1-D Obstacles

(16)

0 0 0

0

sin sin

sin

- 0

0

2 2 2 0

0 2

0

0 0

- F(sin )

sin sin

sin( sin ) 2

sin

sin ( sin ) - intensity~ F(sin ) 4

( sin )

- 2X (sin ) 2 /

- the wider slit,

the narrow the diffracti

X ikX ikX

ikx

X

e e e

ik ik

X kX

kX

X kX

kX kX

θ θ

θ θ θ θ

θ

θ π

⎡ ⎤ − −

= ⎢ ⎥ =

⎣ ⎦

=

→ Δ =

on pattern - secondary peak-very rapidly weak

Diffraction by 1-D Obstacles

(17)

http://micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/

Single-Slit Diffraction

sin , for minima sin

d m

d

δ ≈ θ θ = λ

(18)

Single-Slit Diffraction

- sin

for minima sin , m=1,2,3, d

m d

δ θ

θ λ

≈

= ⋅⋅⋅

(19)

Single-Slit Diffraction

http://www.phys.hawaii.edu/~teb/optics/java/slitdiffr/

(20)

D. Halliday, Fundamentals of Physics

(21)

Convolution

- ( ) ( ) * ( ) ( ) ( ) ( ) ( )

- integrand is a function of and

integration is taken over function of - ( ) vs. ( ) or ( ) vs. ( ) for 1-D r

all r all r

c u f r g r f r g u r d r f u r g r d r

u r

r u

g r g u r g x g u x

= = − = −

→

− −

∫ ∫

- -

- 0

0 0

eflection+displacement - example

( ): function, ( ): arbitrary function

c(u)= ( ) ( - ) ( ) ( - )

( ) ( - )

o

f x g x

f x g u x dx x x g u x dx x x g u x dx

g u x g u x δ

δ δ

∞ ∞

∞

= +

+ −

= + +

∫ ∫

∫

http://www.jhu.edu/~signals/convolve/index.html

(22)

inversion in 3-D

(23)

(24)

(25)

(26)

http://eent3.sbu.ac.uk/staff/baoyb/foct

(27)

Diffraction by 1-D Obstacles

0 0 0

0 0

0 0 0 0

0 0

- two wide slit

slits , each of width 2 , centered at - and

- ( ) 0 - -( )

1 - ( ) -( ) 0 - ( ) ( ) 1 ( )

X x x x x

f x if x x X

if x X x x X

if x X x

= =

= ∞ < < +

+ < < −

− < < −

− < ₀ ₀

0 0

( )

0 ( )

- ( ) is convolution of one wide slit and two narrow slit

( ) ( ) * ( )

x X

if x X x

f x

f two wide slit f one wide slit f two narrow slit

< + + < < ∞

=

(28)

Diffraction by 1-D Obstacles

- Fourier transform

( ) [ ( ) * ( )]

- Fourier transform of a convolution is the product of the individual Fourier transforms

( ) (

Tf two wide slit T f one wide slit f two narrow slit

Tf two wide slit Tf one wide

=

0 0

0

) ( )]

- one wide slit

sin( sin ) (sin ) 2

sin - two narrow slit

(sin ) 2 cos( sin )

slit Tf two narrow slit

F X kX

kX

F kx

θ θ

θ

θ θ

⋅

=

(29)

0

0 0

0

2 2 2 0 2

0 2 0

0

- two wide slit

sin( sin )

(sin ) 4 cos( sin )

sin -intensity

sin ( sin )

(sin ) 16 cos ( sin )

( sin )

F X kX kx

kX

F X kX kx

kX

θ θ θ

θ

θ θ θ

θ

=

Diffraction by 1-D Obstacles

(30)

2 2

2

0 1

1 0

2

0 2

- two wide slit

form of diffraction pattern is a series of cos fringes modulated by (sin / )

- cos function- first zero sin / 2 sin /(2 )

- (sin / ) function- first zero sin kx

kx

kX α α

θ π θ π

α α θ

=

2 0

0 0 2 1

sin /( )

- x

kX X

π θ π

θ θ

=

→ >

Diffraction by 1-D Obstacles

(31)

- three wide slit

- amplitude function

( ) ( ) * ( )

( ) [ ( ) * ( )]

( ) ( ) * (

f three wide slit f one wide slit f three narrow slits

Tf threeo wide slit T f one wide slit f three narrow slits Tf threeo wide slit Tf one wide slit Tf

=

0

0 0

0

) sin( sin )

- (sin ) 2 [1 2 cos( sin )]

sin

three narrow slits

F X kX kx

kX

θ θ θ

= θ +

Diffraction by 1-D Obstacles

(32)

2 2 2 0 2

0 2 0

0

- three wide slits - intensity

sin ( sin )

(sin ) 4 [1 2 cos( sin )]

( sin )

F X kX kx

kX

θ θ θ

= θ +

Diffraction by 1-D Obstacles

(33)

0

- wide slits

width of 2X , centered on a function - amplitude function

( ) ( ) * ( )

( ) [ ( ) * ( )]

( )

N

f N wide slit f one wide slit f N narrow slits Tf N wide slit T f one wide slit f N narrow slits Tf N wide slit T

δ

=

= f one wide slit( ) *Tf N narrow slits( )

Diffraction by 1-D Obstacles

(34)

2 0

2 2 2 0

0 2

0

sin sin

2 sin sin

2

- N wide slits - intensity

sin ( sin ) (sin ) 4

( sin )

Nkx kx

F X kX

kX

θ θ

= θ

Diffraction by 1-D Obstacles

(35)

http://www.msm.cam.ac.uk/doitpoms/tlplib/diffraction/index.php

Experiments

(36)

(b) s = 3 μm (a) s = 12 μm

Slit spacing s and slit width w

(37)

Diffraction patterns from gratings (a) and (b).

(38)

(39)

(40)

The zebra

(41)

- infinite wide slits - amplitude function

( ) ( ) * ( )

( ) [ ( ) * ( )]

( ) ( ) * (

f wide slit f one wide slit f narrow slits Tf wide slit T f one wide slit f narrow slits Tf wide slit Tf one wide slit Tf narrow slit

∞ = ∞

∞ = ∞ s)

Diffraction by 1-D Obstacles

(42)

2 2

2 2 0

0 2

0 0

- infinite number of wide slits - intensity

sin ( sin ) 2

(sin ) 4 (sin )

( sin )

n

kX n

F X

kX kx

θ π

θ δ θ

θ

=∞

=−∞

=

∑

−

Diffraction by 1-D Obstacles

(43)

- significance of the diffraction pattern

Diffraction by 1-D Obstacles

(44)

- narrow slits

1. As the number of slits increases, the main peaks become sharper and narrower, whilest the subsidiary peaks

become rapidly less intense

2. The position and separation of the mai

0

n peaks is constant independent of the number of peaks. The man peaks are separated by an angular deflection given by

(sin ) 2

The distance is determined soly by and separation of a

kx

x θ π

λ

Δ =

ny two neighboring narrow slits.

Diffraction by 1-D Obstacles

(45)

- narrow slits

- The position of the main peak in a diffraction pattern is determined solely by the lattice spacing in an obstacle.

- The shape of the main peak is determined by the overall shape of the obstacle.

Diffraction by 1-D Obstacles

(46)

(47)

- wide slits

- The effect of the motif (one wide slit) is to alter the intensity of each main peak, but the position of the main peaks are unchanged

- intensity envolope → structure of motif

Diffraction by 1-D Obstacles

(48)

- another way of looking at N wide slits - shape function

is zero everywhere outside an obstacle

corresponds to the macroscopic shape of the obstacle within the obstacle

Diffraction by 1-D Obstacles

(49)

- finite lattice

( ) ( ) ( )

( ) ( ) * ( )

( ) ( ) *[ ( ) ( )]

- diffraction pattern

f finite lattice f infinite lattice f shape function f obstacle f motif f finite lattice

f obstacle f motif f infinite lattice f shape function

=

i

(sin ) ( )

{ ( ) *[ ( ) ( )]}

( ) [ ( ) ( )]

( ) [ ( ) * ( )]

F Tf obstacle

T f motif f infinite lattice f shape function Tf motif T f infinite lattice f shape function Tf motif Tf infinite lattice Tf shape function

θ =

=

i

i i

i

Diffraction by 1-D Obstacles

(50)

- N wide slits

( ) ( ) * ( )

( ) ( ) *

[ ( ) ( )]

f N wide slits f one wide slit f N narrow slits f N wide slits f one wide slit

f narrow slits f shape function

=

∞ i

Diffraction by 1-D Obstacles

(51)

- diffraction pattern

( ) { ( ) *

[ ( ) ( )]}

( ) [ ( ) ( )]

Tf N wide slits T f one wide slit

f narrow slits f shape function Tf one wide slit T f narrow slits f shape function

=

∞

= ∞

i

i i

( ) ( ) * ( )

- three types of structural information that concerning the lattice

that concerning the motif

that concerning the shape of the entir

Tf one wide slit Tf narrow slits Tf shape function

= i ∞

e crystal

Diffraction by 1-D Obstacles

(52)

Diffraction by 1-D Obstacles

(53)

Diffraction by 1-D Obstacles

(54)

Summary

- the of the main peaks gives information on the

- the of each main peak gives information on the overall object

position

shape

set of in

- the of all th

lat

e m

tens ain

tice

peak

iti s g

s

i hape

es ves information

on the structure of the motif

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

Chapter 11 Diffraction

Reading Assignment: