Cherenkov radiation

nv

cc cos

真空中匀速直线运动带电粒子不辐射？

带电粒子在介质中运动产生诱导电流，当粒子速度超过介质内光速时，激发次波与原粒子电磁场干涉，可以形成辐射场。方向性好 :

Cherenkov1934,Frank &Tam 19371958 Nobel prize

30 ( , )( , )

4

/ ,V

j t xA t x d x

r

where t t r c r x x

dtetxAA

deAtxA

ti

ti

),(2

1

),(

介质中运动电荷产生的场可用推迟势计算（用介质光速 c’ ）

)(),( xxvetxj

讨论

特定频率分量

)cos(sin4 3

0

c

n

vR

e

c

neiAn

c

niAkiB

ikR

xdeR

e

c

e

tdtveR

e

c

e

xc

n

viikR

tdvx

c

xntiikR

xnRr

)cos1

(

20

2

)(

20

2

8

)(8

Bnvcncv

allforradiationnoncvif

,/cos,/

,/

)/(3203

20 ),(

8),/(

8

),(2

1

crti

V

ti

V

ti

extjxdtdR

excrtjxddtR

dtetxAA

22302 4

B

n

RcdtRnS

d

dW

BncE

Angular distribution of the radiation energy

2

2 2

0

( ) ( ) ( )

2 2 | | 4 | |

i t i tE t dt E t dt E e d E d E t e dt

E E d E d E d

where

韧致辐射和同步辐射的频谱

22

2

20

2

14 nv

c

c

e

dL

dW

2

22 )(

v

cn

21( cos ) 02 cos 2 cos

ni x iv c n ne dx e dx L

v c v c

由典型的色散曲线（ show ）知Cherenkov 辐射只包括某一频段

单位频率间隔单位路程的辐射能量角分布2 2 2

2 3 2 20

1 ( cos )8

dW e c n

d dL c v n v c

单位频率间隔单位路程的辐射能量

通过测量辐射角来确定粒子速度 !

Dispersion (p309)当电磁波入射到介质内时，由束缚电子散射的次波会叠加成介质内传播的电磁波。宏观电磁现象由极化强度 P 和 磁化强度 M 决定。

束缚电子（谐振子）散射 20 0

i tex x x E e

m

02 20

( )0 2 22 2 2 2 2

00

1

1, tan

( )

i t

i t

ex E e

m i

eE e

m

sin4

)(4

20

)ˆˆ(

20

0

rc

xeE

xnnrc

eE

EE

散射波

15 70

2 280

30

24 10 ( , 5 10 )

106

k cm

m

e

mc

平均能流密度4 2 4

202 3 2 2 2 2 2 2 2

0 0

sin32 ( )

e ES

c m r

2 4 2

2 2 2 2 2 20 0

8, 2.8

3 ( ) 4e

e

r er fm

c m

散射截面

0

0

0 Rayleighre ,3

84

0

2

)Thomson(,3

8 2

electronfreere

22

0

8, ,

3er resonant

稀薄气体近似：忽略分子间相互作用， 单位体积电子数 N ，利用束缚电子散射结果

2

2 20

2

0 2 20

1

1

NeP Nex E

m i

Ne

m i

2 220

2 2 2 2 20 0

2

2 2 2 2 20 0

( ) 1( )

( ) ,( )

r

r

Nereal

m

Neimag

m

，色散

吸收

通常测定的折射率即为实部 nr n i

2 220

2 2 2 2 20 0

2

2 2 2 2 20 0

12 ( )

2 ( )

Nen

m

Ne

m

2

0 2 2i

i i i

fNe

m i

考虑到多个固有频率（激发态），分支为 fi

2 222 2

2 2 2 2 20

2

2 2 2 2 20

( )1

( )

2 ( )

i i

i i i

i i

i i i

fNen

m

fNen

m

Scattering and Diffraction (ch10, p456)

Involved scales: Wavelength and size of target

L

L

L

Lowest order induced EM multipoles oscillate and radiate energy

Need more systematic treatment with multipoles

Semi-geometric methods

00 0

0 0/

ikn xinc

inc inc

E E e

H n E Z

2

0

0

1[( ) / ]

4

/

ikr

SC

SC SC

eE k n p n n m c

r

H n E Z

The incident fields are

Induced dipole moments (p and m) radiate energy in all directions. The scattered (radiated) fields (in the direction n) are (Eq 10.2)

The differential cross section = power radiated per unit solid angle, per unit incident flux

2 * 2

00 0

* 20

0

4* * 2 4

20 0

1| |

2( , ; , )

1| |

2

| ( ) / |(4 )

SC

inc

r EZd

n nd E

Z

kp n m c

E

Rayleigh’s law: universal characteristic of the of the long wave length scattering by any finite system (dipole scattering)

Scattering by a small dielectric sphere of radius a

The electric dipole moment is (4.56) at p158

30

14 , 0

2r

incr

p a E m

2

4 6 * 20 0 0

1( , ; , ) | |

2r

r

dn n k a

d

So the differential cross section

The incident wave is unpolarized, the parallel and perpendicular components (w.r.t. the scattering plane) are

2 24 6 4 621 1

cos ,2 2 2 2

r r

r r

d dk a k a

d d

2

2

2

4 6 2

2

4 6

/ / sin( )

/ / 1 cos

1 1(1 cos )

2 2

18

3 2

r

r

r

r

d d d d

d d d d

dk a

d

dd k a

d

The polarization, differential and total scattering cross section are (see Fig 10.2 at p459)

Scattering by a small perfectly conducting sphere of radius a

The electric dipole moment is (see section 2.5 at p64)

30

3

4 ,

2

inc

inc

p a E

m a H

4 6 * * 20 0 0 0 0

1( , ; , ) | ( ) ( ) |

2

dn n k a n n

d

So the differential cross section

4 6 2

2

2

5[ (1 cos ) cos ]8

3sin( )

5(1 cos ) 8cos

dk a

d

The differential cross section and polarization

The cross section has a strong backward peaking caused by electric dipole -- magnetic dipole interference. The polarization reaches 1 at 60 degrees and is positive through the whole angular range.

Perturbation theory- the medium is supposed to have small changes in its response to applied fields

22

0 0 0 0 02( ) ( )

DD D E B H

t t

2 20 0 0( ) ( ) ( )k D D E i B H

The wave equation for D

With harmonic time variation, the above equation becomes

A formal solution is| |

(0) 30 0 0| |

(0)

1[ ( ) ( )]

4

ik x x

x x

ikr

SC

eD D d x D E i B H

eD A

r

30 0 0

3 00 0

1[ ( ) ( )]

41

{[ ( )] ( )}4

ikn xSC

ikn x

A d x e D E i B H

d xe n D E n n B Hk

The scattering amplitude

The differential cross section (a formal solution)

* 2

(0) 2

| |

| |SCAd

d D

0 0( , )D E B H

Born approximation (0)0

0

(0)0

0

( )

( )

D E D x

B H B x

0(0) (0) (0)00 0 0

0

,ikn xD D e B n D

* 23 * *

0 0 00 0 0

[ ( ) ( ) ]4

iq xSCA kd xe n n

D

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

The unperturbed fields

so

Suppose that the scattering region is a uniform dielectric sphere of radius a, is constant inside a sphere and vanishes outside

*2 *

0 30 0

sin cos( )SCA qa qa qa

kD q

AAAAAAAAAAAAAAAAAAAAAAAAAAAA

At low frequencies or in the forward direction

Perform the integral

2

4 6 * 20

00

lim ( )3q

Born

dk a

d

Blue sky

The effective variation in dielectric constant is

where is molecular polarizability (p161)

If individual molecules are assumed to possess dipole moments 0 ( )j mol jp E x

mol

24

2 * 202

( )16

jiq x

molj

d ke

d

0 ( )mol jj

x x

The differential cross section is

The total cross section

For dilute gases 1r molN

4 42 2

2 2

2| 1| | 1|

6 3r

k kn

N N

42

2

2| 1|

3

kN n

N

0( ) xI x I e

In traversing a thickness dx of the gases, the fractional loss of flux is so the beam intensity is

N dx

with absorption or attenuation coefficient

Discussion (p467)

1. Light received away from the incident beam is more heavily weighted in high-frequency (blue) components than the spectral distribution of the incident beam

2. Transmitted beam becomes increasingly red in its spectral composition, as well as diminishing in overall intensity

3. The blueness of the sky, the redness of the sunset, the waneness of the winter sun, and the ease of sunburning at midday in summer

4. Relative intensities: Zenith Sunrise-Sunset

Red (6500A) 0.96 0.21

Green (5200A) 0.90 0.024

Violet (4100A) 0.76 0.000065