Chapter 10 Diffraction

Reading Assignment:1. D. Sherwood, Crystals, X-rays, and

Proteins-chapter 6http://www.matter.org.uk/diffraction/Default.htm

Contents

Diffraction and Information

Mathematics of Diffraction

Fourier Transform

3

1

2

4

Diffraction

5 Experimental Limitations

Interaction of waves with obstacles- interaction of waves with obstacles

infinite plane wave with wave vector and frequency

obstacle- perturb the wave motion in some way

-scattering- wave-obstacle interaction such that the

dimensions of obstacles and wavelength are comparable

diffraction- wave-obstacle interaction such that the

dimensions of obstacles are much larger than the

wavelength of the wave motion

( )i k r wtoeψ ψ −=

k

w

X-ray (~1Å)

crystal (~10mm)

atom (~1nm) scattering-microscopic

diffraction-macroscopic

Diffraction of X-rays

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

Diffraction of Water Waves

- ripple tankrule-periodic dipping plane water wave

long-no reflection

- transparent base-light shadow on the screen

- synchronize light withwave motion- frozen in stroboscopic ripple

tank

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

Diffraction of Water Waves

- obstacle- plane wavecurved wave

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

D. Halliday, Fundamentals of Physics

Diffraction of Water Waves

- when slit width is wide compared to the wavelengthof the wave motion, the diffraction effects are masked

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

Diffraction and Information

- plane wave obstacle semi-circulardiffracted wave contains additional

information about obstacle

- When a wave interacts with an obstacle, diffractionoccurs. The detailed behavior depends solely on thediffracting obstacles, and so the diffracted waves maybe regarded as containing information on the structureof the obstacles.

- ex) X-ray

crystal

diffracted pattern

information about nature of moleculesand the way they are packed

reconstructobstacle

Diffraction of Light

D. Halliday, Fundamentals of Physics

Poisson’s Bright Spot

http://www.richmond.edu/~ebunn/phys205/cache/poisson.html

Diffraction of X-rays

crystal: well-defined, 3D orderedarray

X-ray

atom, molecule

well-defined phase relationshipbetween scattered waves

clear and precisediffraction pattern

* liquid- diffuse and impreciseD. Sherwood, Crystals, X-rays, and Proteins

Mathematics of Diffraction

-assumption-structure and all physical properties ofthe diffracting obstacle are known.

(diffraction pattern-recipe)-plane electromagnetic waves of a single

frequency and wavelength are incident normally on the obstacle.

(a) frequency and wavelength

w λ

incident wave

diffracted wave

E w w

w

electrons in the moleculeoscillate

free

Mathematics of Diffraction

(a) frequency and wavelength

in any medium, velocity of electromagnetic radiationis constantwavelength is also unchanged

* diffracted waves have the same frequency andwavelength as the incident waves.

* electrons in the molecules are not really free

* during one cycle of the X-radiation, the electron has hardly changed its position relative to thenucleus electron forget the presence of nucleus electrons behave as if it were free

λ

19

16

X-ray : 0.1 nm, 10 Hzorbital motion of an electron: 10 Hz

ww

λ = =

=

Mathematics of Diffraction

(b) spatial consideration

frequency and wavelength unchangedsome other effect

plane wave spatially different wave* information of the wave is somehow contained in the

way in which they are spread out in space.

emit in all directions

superpositionemit in all directionsin space

emit in all directions

Mathematics of Diffraction

(c) significance of the wave vector k

( )o=

2magnitude :

direction: normal to the advancing wavefront

spatial information* total set of diffracted waves may be represented by a set of wave vectors all of which have th

i k r wte

k k

k

ψ ψπλ

−

= =

→

e same magnitude, equal to that of incident wave, but different directions

Mathematics of Diffraction(d) mathematical description

- assumption- structure and physical properties known- phenomenon- (1) waves are propagated through obstacle (2) obstacle perturb these waves

- arbitrary origin, position vector

11 1

1 1

( )

,

infinitesimal volume element centered on by

- volume element centered on

perturbing effect ( )

i k r wt

r

r d r

d r r

f r d r

e −→

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

Mathematics of Diffraction

1

1

1

1 1

1

1 1

1 1

( )

( )

( )

- two different way of combining and ( )

diffracted wave from volume element

( ) or

( )

- consider a pe

-

-

i k r wt

i k r wt

i k r wt

f r d r

dr

f r d r

f r d r

e

e

e

−

−

−

+

11 1 1

2

( )

rfectly absorbing obstacle ( ) 0 no diffracted wave second equation is correct

- wave diffracted from volume element ( )

- wave diffracted from volume element (

i k r wt

f r

d r f r d r

dr f r

e −

→ =→

=

=

22 1

( )) i k r wt d re −

Mathematics of Diffraction

1 2

3

1 21 1

3 3

( ) ( )

( )

- Principle of superposition

diffraction pattern = ( ) ( )

( )

i k r wt i k r wt

i k r wt

f r d r f r d r

f r d r

e e

e

− −

−

+

+

V

-

19 19

( ) = ( )

= ( )

- X-ray: ~0.1 nm, ~ 10 Hz, one cycle ~ 10 sec- no recording device is availabele- diffraction expe

iwt

V

i k r wt

ik r

f r d r

e f r d r

w E

e

e

λ −

−∫

∫

riment- time average of the intensity of the diffracted wave

Mathematics of Diffraction

- diffraction pattern = ( )

- limits on the integral

( ) : finite mathematically not well behaved

define ( ) over an infinite range dividing the values into two domains

diff

V

ik rf r d r

f r

f r

e

→

∫

all

raction pattern = ( )

( )

( ) : amplitude function- diffraction pattern of any obstacle is the Fourier transform of the ampli

Fourier

tude

trn

funct

asform of ( )

ion

ik r

r

f r

f

d r

f r

r

e∫

Significance of Fourier Transform

all

- diffraction pattern = ( )

1. integrand - function of and , integration - over

diffraction pattern ( )

information in the diffraction pattern-spatial k

2.

ik r

r

ik

f r d r

k r r

F k

e

e

→

←

∫

: wave, ( ) : amplitude type function passive scatterer replaced by active source identical

r f r→ →

Fourier Transform and Shape

1 2

1 1

0 0 1 1 2 2

- a linear array of scattering center in an obstacle- phase at point 0=0

2 phase at point 1= OA( cos )

- total scattered wave= ( ) ( ) ( )ik r ik r

r k r

f r d r f r e dr f r e dr

π θλ

= =

+ + +

= ( )

- exponential term-specifiy phase relationship between scattering centers and an arbitrary origin

obstacle

ik rf r e d r∫

Fourier Transform and Information

- ( ) ik rf r e

amplitude modulated

5

- AM radio radio frequency (of order 10 Hz) audio frequency (of order 500 Hz)

ik re ( )f r

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

Fourier Transform and Information

- diffraction pattern in terms of Fourier transform

(i) the description is, as required, a function of k (ii) the decription agrees with our physical knowledge of the interaction of wave w

ith obstacles (iii) the description represents a general solution of three-dimensional wave equation (iv) the description is capable of containing the required information

Inverse Transform

all

all

- diffraction pattern ( )

( )= ( )

( ) is Fourier transform of ( )- inverse transform

( )= ( )

- ( ) : diffraction pattern functio

ik r

r

ik r

k

F k

F k f r d r

F k f r

f r F k dk

F k

e

e−

∫

∫

2

n- information on spatial distribution of diffraction pattern

( ): information on the structure of obstacle

- diffraction experiment intensity ( ) ( ) ( )

f r

F k F k f r→ ∝ → →

Microscope

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

Experimental Limitation

all

- diffraction pattern ( )

( )= ( )

- information contained in all space physically impossible to scan all space to collect some information lost- phase problem measur

ik r

r

F k

F k f r d re∫

2

e only the intensity not the complex amplitude of diffraction pattern

( ) ( )

observe ( ) lost information on

iF k F k e

F k

δ

δ

=

→