due to the oscillating charge in the antenna

Along this line onedoes not observeany acceleration

Radiation from a dipole-antenna

R

charge ; ( , ) ( / )rad radE q E R t observed acceleration t R c

20

1 1( , ) ( / ) ;

4rad

eE R t acc t R c

R c

To get the dimension right !

Electromagnetic radiation( / )

0

( / )0

mass acceleration force=charge field

[ ] ( / ) [ ]

[ ]

[ ] ( )

i t R cin

i t i c R

i kRin

m acc e E t R c e E e

e E e e

e E t e

0

25

0 20

( ) ;

( )

1 2.82 10

4

i kRrad

in

E t er

E t R

er Ang

mc

Here one observes the full acceleration,but delayed in time by R/c !

1( )radE R

R

Guess that

2 22 2

1 14radE energy density radiated energy R

R R

20

2

2

m/sec Force = Energy

1( , )

4

OK(m/sec)

rad

e acce E R t

R c

2

Polarisation

P=cos2(ψ)

P=1

Interference (mathematical)

z

r

rk zkz

in

2

k’ Q

rk ' outrQrkk )' ( res

0. . (1 )isc ampl r e Q r

0. . ( ) isc ampl r e d Q rr r

Number density

0. . ji

j

sc ampl r e Q r

many

wavecrest

k1

2

as drawn #2 is behind #1 for ”in”

but ahead for ”out” , therefore ” – ”

Phase

0 .

1,2 … many

The dream experiment

2 *2 1 2 2 2 2;i iA f e f e I A A 1Q r Q r

2 2

1 2 1 2

2 1 2 1 2

( ) ( )2 21 2 1 2 1 2

{ } { }

i i i i

i i

I f e f e f e f e

f f f f e f f e

1 1Q r Q r Q r Q r

Q r r Q r r

2 2 122 1 2 1 2.

12

sin( )2

orient average

Q rI f f f f

Q r

2

.

sin( )2 i j

i i jorient averagei i j i j

Q rI f f f

Q r

1,2 … many

DetectorViewingField

Detector

X-ray beam

Kr

Gas cell

sin2

)(2sin

1 2

kQ

QfIntensity

Measuring atomic and molecular formfactors from gas scattering

j

ijmol

ielatom

jeff

def

rQ

rQ rr)(

a=[15.2354 6.70060 4.35910 2.96230]; b=[3.0669 0.241200 10.7805 61.4135];c=1.71890; % Ga

a=[16.6723 6.0710 3.4313 4.27790];b=[2.63450 0.2647 12.9479 47.7972];c=2.531; % As

24( / 4 )0

1

( ) jb Q

j jj

f Q a e c

1cos 2 ( )

sin sin( )

2 2sin

Qri x

i Q r

x Qriorientationalaverage

Qr e dxe d d Qr

eQrd d

Q r

Unit sphere

d

d

r

Q

iorientationalaverage

e Q r

r

sinsind

sind d d

Fourier transform of a Gaussian

Fourier transform : ( ) ( ) i q xF q f x e dx

2 2

( ) a xf x Ae

2 2/(4 )( ) q aAF q e

a

2 2 2 2/ 2 / 21or with ( ) ; ( )

2x qf x e F q e

( ) ( ) when 0Gaussian x x

( ) ( ) (0) f x x dx f

F.T. 1 (delocalize( ) ( )

localized

d in )

in

Gaussi qan x x

x

Convolution of 2 Gaussians

2 2 2 2( ) / 211 1 2/ 2

1( )x xx

h x e e dx

2 2 2 2 2 2

1 2/ 2 / 2 / 2( ) q q qH q e e e

i.e. h(x) is also Gaussian with 2 2 21 2

Side View

Top ViewRing wave (2D) orSpherical wave (3D)

2 23D : 4 independent of i k re

A r rr

Energy density Surface area

locally a plane wave iAe k r

Aperture d Almost planewave when

d

Perfect planewave

k’

k

A point scatterer in the beam

i kReA

R

Spherical wave

2Flux : c Intensity Flux Area scattered in

thru

dI

d

22

2

2

2

i kR i kR Ac e e R

R

c A

dA

d

Defines thescatteringcross section1c

Area is

2R

scattering length

scattI i k x

in e

2LL=(N+1)(

2LL=NNo real beam is perfectly monochromatic

From the 2 equations, derive

21

2LL

the longitudinal coherence length

P(

wavelength band

No real beam is perfectly collimated

P(

D

2 TL

LT

Hereand B beams in phase

A and B out-of-phase

show from the figure that

2T

RL

D

With R being the distance from to

source observation point

A

B The transversecoherence length

Absorption

zeIzI

dzzIdI

0)(

)(

) ,number atomic( Z

34 1

) ,(

ZZ

The experimental setup

X-rays

Sample

Rotation

Scintillator

20m

CCD

< 2 mdeg

E = 8-27 keV

0.7 x 0.7 mm

Tomography

• Study the bulk structures, 3D

• Nondestructive• Small

lengthscales (350 nm)

Fra http://www.unge-forskere.dk/

Galathea III

Single slice

100 microns

Compton Scattering

3

'1 (1 cos )

'

3.86 10

C

C

k

Åmc

Energy and momentum conservation