Chapter 10 Diffraction

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Chapter 10 Diffraction

Reading Assignment:

1. D. Sherwood, Crystals, X-rays, and Proteins-chapter 6

http://www.matter.org.uk/diffraction/Default.htm

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Contents

Diffraction and Information Diffraction and Information

Mathematics of Diffraction Mathematics of Diffraction

Fourier Transform Fourier Transform 3

1 2

4

Diffraction Diffraction

5 Experimental LimitationsExperimental Limitations

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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 wt

ψ = e

ⁱ ⁻

k w

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X-ray (~1Å)

crystal (~10mm)

atom (~1nm) scattering-microscopic

diffraction-macroscopic

Diffraction of X-rays

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

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Diffraction of Water Waves

- ripple tank

rule-periodic dipping Æ plane water wave

long-no reflection

- transparent base-light Æ shadow on the screen - synchronize light with

wave motion- frozen in Æ stroboscopic ripple

tank

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Diffraction of Water Waves

- obstacle- plane waveÆcurved wave

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D. Halliday, Fundamentals of Physics

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Diffraction of Water Waves

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

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Diffraction and Information

- plane wave Æ obstacle Æ semi-circular

diffracted wave contains additional information about obstacle

- When a wave interacts with an obstacle, diffraction occurs. The detailed behavior depends solely on the diffracting obstacles, and so the diffracted waves may be regarded as containing information on the structure of the obstacles.

- ex)

X-ray

crystal

diffracted pattern

information about nature of molecules and the way they are packed

reconstruct obstacle

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Diffraction of Light

D. Halliday, Fundamentals of Physics

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Poisson’s Bright Spot

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

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Diffraction of X-rays

crystal: well-defined, 3D ordered array

X-ray

atom, molecule

well-defined phase relationship between scattered waves

clear and precise diffraction pattern

* liquid- diffuse and imprecise

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Mathematics of Diffraction

-assumption-structure and all physical properties of the 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 molecule oscillate

free

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Mathematics of Diffraction

(a) frequency and wavelength

in any medium, velocity of electromagnetic radiation is constantÆwavelength is also unchanged

* diffracted waves have the same frequency and wavelength 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 the

nucleusÆ electron forget the presence of nucleus Æ electrons behave as if it were free

λ

19

16

X-ray : 0.1 nm, 10 Hz

orbital motion of an electron: 10 Hz w

w

λ = =

=

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Mathematics of Diffraction

(b) spatial consideration

frequency and wavelength unchanged some other effect

plane wave Æ 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

superposition

emit in all directions in space

emit in all directions

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Mathematics of Diffraction

(c) significance of the wave vector k

( )

= o

magnitude : 2

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 wt

e

k k

k

ψ ψ

π λ

−

= =

→

i

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

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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

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

⁻

→ ⁱ

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Mathematics of Diffraction

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 d r

f r d r

f r d r

e

e e

−

+

i

i i

1 1 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

d r f r

e

⁻

→ =

→

=

i

2 2 1

( )

)

e

^{i k r}ⁱ ⁻^wt d r

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Mathematics of Diffraction

1 2

3

1 1 2 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

− −

−

+ +

i i

i iii

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

i k r

f r d r

e f r d r

w E

e

λ ⁻

∫

−

∫

i

riment- time average of the intensity of the diffracted wave

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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

i k r

f r d r

f r

e

→

∫

ⁱ

all

raction pattern = ( )

( )

( ) : amplitude function

- diffraction pattern of any obstacle is the Fourier transform of the ampli

Fourier

tude trn

funct asform of ( )

ion

i k r r

f r f

d r

f r

r

∫ e i

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Significance of Fourier Transform

all

- diffraction pattern = ( )

1. integrand - function of and , integration - over diffraction pattern ( )

information in the diffraction pattern-spatial k 2.

i k r r

i k

f r d r

k r r

F k

e

→

←

∫

ⁱ

: wave, ( ) : amplitude type function

passive scatterer replaced by active source identical

r f r

→ →

i

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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

phase at point 1= 2 OA( cos )

- total scattered wave= ( ) ( ) ^{i k r} ( ) ^{i k r}

r k r

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

π θ

λ ⁼ ⁼

+ ⁱ + ⁱ +

i

= ( )

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

obstacle

i k r

f r e d r

∫

ⁱ

ii

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Fourier Transform and Information

- ( )f r

e

^{i k r}ⁱ

amplitude modulated

5

- AM radio

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

i k r

e

ⁱ ^{f r}^{( )}

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Fourier Transform and Information

- diffraction pattern in terms of Fourier transform

(i) the description is, sa required, a function of k

(ii) the decription agrees with our physical knowledge of the interaction of wave with obstacles

(iii) the description represents a general solution of three-dimensional wave equation

(iv) the description is capable of containing the required information

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Inverse Transform

all

- diffraction pattern ( ) ( )= ( )

( ) is Fourier transform of ( ) - inverse transform

( )= ( )

- ( ) : diffraction pattern functio

i k r r

i k r k

F k

F k f r d r

F k f r

f r F k d k

F k

e

⁻

∫

i

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

→ ∝ → →

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Microscope

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Experimental Limitation

all

- diffraction pattern ( ) ( )= ( )

- information contained in all space

physically impossible to scan all space to collect some information lost

- phase problem measur

i k r r

F k F k

∫

f r

e

ⁱ d r

2

e only the intensity not the complex amplitude of diffraction pattern

( ) ( )

observe ( ) lost information on F k F k ei

F k

δ

=

→