光散乱の古典理論

• 光散乱の古典理論–誘電率の揺らぎによる光散乱

• 液体の光散乱–空間相関と構造因子

臨界タンパク光臨界タンパク光混合液体系(Binary Liquid system)の例

２成分の界面

温度計

Methanol-rich phase

Hexane-rich phase

Hexane（Ｃ６Ｈ１４） ： Methanol（ＣＨ３ＯＨ）

= 6 : 4 (in mol ) ~ 5 : 1 (in volume)

臨界タンパク光臨界タンパク光 -- OpalescenceOpalescence光の波長程度の粒径に液滴が成長したときに、強く光が散光の波長程度の粒径に液滴が成長したときに、強く光が散乱される。乱される。

Partially Miscible Binary SystemsPartially Miscible Binary Systemshexane and methanolhexane and methanol

Begin with pure A (left side of graph). Begin with pure A (left side of graph). Only have one phase.Only have one phase.As B is added:As B is added:

Below the saturation limit, there Below the saturation limit, there will only be one phase will only be one phase Above the saturation limit, there Above the saturation limit, there will be two phases.will be two phases.

In this diagram, the composition of In this diagram, the composition of one of the phases is aone of the phases is a’’ and the and the composition of the other phase is composition of the other phase is aa””..

Eventually enough B is added such that Eventually enough B is added such that A is actually dissolved in B and you A is actually dissolved in B and you once again only have one phase.once again only have one phase.Above Above TTucuc (the upper critical (the upper critical temperature), the two liquids are temperature), the two liquids are miscible in all proportions.miscible in all proportions.

No phase separation occurs. No phase separation occurs. Mole fraction of methanol,

風景の変化風景の変化

Q: Does aerosol have anything to do with the difference?

ステンドガラスステンドガラス金属・半導体微粒子の分散による光吸収・散乱金属・半導体微粒子の分散による光吸収・散乱

火星の夕焼け火星の夕焼け

NASANASA MARS PATHFINDERMARS PATHFINDER

粒径による散乱の違い粒径による散乱の違い

粒径粒径 a a 光の波長光の波長 λλa a>> λλ

Mie-scatteringExact solution of scattering by spheres similar in size

to wavelength of radiation results in larger scattering in the forward direction.

Geometric scattering

Coherent vs. Incoherent light scatteringCoherent vs. Incoherent light scatteringCoherentCoherent light scattering: scattered wavelets have light scattering: scattered wavelets have nonrandom nonrandom relative phases in the direction of interest.relative phases in the direction of interest.

IncoherentIncoherent light scattering: scattered wavelets have light scattering: scattered wavelets have randomrandom relative relative phases in the direction of interest.phases in the direction of interest.Example:Example:

Forward scattering is coherent—even if the scatterers are randomly arranged in space.

Path lengths are equal.

Off-axis scattering is incoherentwhen the scatterers are randomly arranged in space.

Path lengths are random.

Coherent scattering:Coherent scattering:

Total complex amplitude, . Irradiance,Total complex amplitude, . Irradiance, II∝∝ µµ AA22.. So:So: IIcc∝∝µµ NN22

Incoherent scattering:Incoherent scattering: Total complex amplitude, Total complex amplitude,

The irradiance The irradiance

So So incoherent incoherent scattering is weaker than scattering is weaker than coherent coherent scattering, scattering, but not zerobut not zero..

Coherent vs. Incoherent ScatteringCoherent vs. Incoherent Scattering

11

N

cohm

A N=

∝ =∑

2

1 1 1

1 1 1 1

exp( ) exp( ) exp( )

exp[ ( )] exp[ ( )]

N N N

m m nm m n

N N N N

m n m nm n m nm n m n

I i i i

i i N

θ θ θ

θ θ θ θ

= = =

= = = == ≠

∝ = −

⎛ ⎞ ⎛ ⎞= − + − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑ ∑

∑∑ ∑∑

1exp( )

N

incoh mm

A iθ=

∝∑

θm=θn m nθ θ≠

The equations of optics are Maxwell’s equations.

where is the electric field, is the magnetic field, ρ is the charge density, ε is the permittivity, and µ is the permeability of the medium.

/

0

BE EtEB Bt

∂ρ ε∂∂µε∂

∇ ⋅ = ∇× = −

∇ ⋅ = ∇× =

rr r r r

rr r r r

Er

Br

)),((),( 0 trItr δεεεε +=

runit vecto : r,unit tenso : inI

)}(exp{),( 0 trkiEntrE iiii ω−=

iik ω ,0 , Eni

rR −

R

D

Or

rd 3

V

IR εεε 0)( =

誘電率の揺らぎによる電磁波の散乱

tDH

tBE

BD

∂∂=×∇

∂∂−=×∇

=⋅∇=⋅∇

00

iiii BHDE , , ,

SSSS BHDE , , ,

Si

Si

Si

Si

BBBHHHDDDEEE

+=+=+=+=

入射波と散乱波

tDH

tBE

BD

ii

ii

i

i

∂∂=×∇

∂∂−=×∇

=⋅∇=⋅∇

00

2

2

0 tDE ii ∂

∂−=×∇×∇ µ

ii ED 0εε=

0=⋅ ii nk

AAA 2)( ∇−⋅∇∇=×∇×∇

022

22 =

∂∂−∇

tE

cE ii

εii c

k ωε=

0th order solution• Maxwell equation

1st order solution (1)• Maxwell equation

2

2

0 tDE SS ∂

∂−=×∇×∇ µ )),((),( 0 trItr δεεεε +=

)(),(

)))(,((),(2

000

0

δδεεεεεεδεεεε

oEtrEEEEtrIEtrD

iSi

Si

+++=

++==

iSS

iSS

EtrDE

EtrED

εδε

εε

δεεεε),(1),(

0

00

−=

+=

}),({ 022

22

iS

S EtrtD

cD δεεε ×∇×−∇=

∂∂−∇

1st order solution (2)• Hertz vector

π×∇×∇=SD

iEtrtc),(02

2

22 δεεπεπ −=

∂∂−∇

∫ −= Vi

rRtrEtrrdtR )',()',(

41),( 03 δεεπ

π

rRc

tt −−= ε'

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×∇×∇= ∫V

iRRS rR

trEtrrdtRD )',()',(41),( 03 δεεπ

1st order solution (3)• Scattering field

Fourier component of the fluctuation δε(r,t)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×∇×∇=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×∇×∇=

∫

∫

−

V

trkiiRR

Vi

RRS

iientrrR

rdE

rRtrEtrrdtRE

)'(3

0

3

):)',((4

)',()',(4

1),(

ωδεπε

δεπε

'),(21)',( tierdtr ωωδεωπ

δε ∫=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

×∇×∇=

−−∫∫ ')(3

0

):),((21

4),(

tiiV

rik

RRS

ii enrderR

rd

EtRE

ωωωδεωπ

πε

1st order solution (4)Far field approximation

)ˆ('

/ˆ ˆ

S

SS

krRc

tt

RRkkrRrR

⋅−−≅

=⋅−=−

ε

)):),((21

(4

),(

)(

)ˆ)((30

Rc

i

iti

V

rkc

iki

RRti

S

i

Siii

enred

erR

rdeEtRE

ωωεω

ωωεω

ωδεωπ

πε

−

−−−

∫

∫

×

−×∇×∇=

)):),((21

(4

),(

)(

)ˆ)((30

Rc

i

iti

V

rkc

iki

RRti

S

i

Siii

enred

erR

rdeEtRE

ωωεω

ωωεω

ωδεωπ

πε

−

−−−

∫

∫

×

−×∇×∇=

1st order solution (5)• Scattering wave

• Slow fluctuation approximation

SiSSSiSiS kkqkck

ck −≡=−≡−≡ ,ˆˆ)( , ωεωωεωωω

)):),(((4

),(3

0iV

iqrRikRR

tiS ntrerR

rdeeEtRE Si δεπε

ω ∫ −×∇×∇=−

ω積分の外に出す。

1st order solution (6)Far field approximation again

Fourier transformation of δε in r-space

order in , 111 -RikRrR ×→×∇→− −−

{ }):),((4

),( 3)(0 iViqr

SStRki

S ntrredkkeREtRE iS δεπε

ω ∫××−= −

∫= Viqr trredtq ),(),( 3 δεδε

SEn ofr unit vecto :S SS

SS nE Ekn 0S ,0 ==⋅

{ }):),((4

),( )(00 iSSStRkiS ntqkkne

REtRE iS δεπε

ω ××⋅−= −

)()()( BACCABCBA ⋅−⋅=××

• 誘電率の揺らぎテンソル

• Scattering amplitude

):),((),( iSSi ntqntq δεδε ⋅=

1st order solution (7)

),(4

),( )(2

00 tqeR

kEtRE SitRkiSS iS δε

πεω−=

RetRE

tRkiS

iS )(

0 ),(ω−

∝

),(),( 00 tqEtRE SiS δε∝

球面波

入射振幅、揺らぎに比例線形応答

0),(0 =tRES

Power spectrum of the scattering wave• Wiener-Khinchin theorem

dtetCqI tiSS S∫∞

∞−

−= ωω )(),(

)0()()()()( *2

00*

SSSSS EtEEtEttEtC =−+=

dssEstET

T

SST ∫ += ∞→ 0* )()(1limL

エルゴード性が成り立つときはアンサンブル平均になる。

tiSiSi

SSS

ieqtqR

kEEtE ωδεδεπε

)0,(),(4

)0()( *22

20

*

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫∞

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= tiSiSiiS eqtqdtcR

IqI ωδεδεωπ

ω )0,(),(8

),( *54

220

等方的媒質による散乱• 誘電率の揺らぎテンソル

),()():),((),( tqnnntqntq iSiSSi δεδεδε ⋅=⋅=

TTT

δεδρρεδε

ρ⎟⎠⎞

⎜⎝⎛

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂=

通常小さい

∫∞

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= ti

TiS

iS eqtqdtnncR

IqI ωδρδρδρδεω

πω )0,(),()(

8),( *

22

5

4

220

密度揺らぎ 温度揺らぎ

∫ −= Viqr trredVqtq )0,0(),()0,(),( 3 δρδρδρδρ

空間フーリエ変換

密度揺らぎの時間空間相関関数

• 動的構造因子

• 構造因子

• 散乱強度

),()(8

),(2

25

4

220 ω

δρδεω

πω qSnn

cRVIqI

TiS

iS ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

動的構造因子

∫ ∫∞

∞−

−=V

tiiqr trdteredqS )0,0(),(),( 3 δρδρω ω

),(2

)( ωπω qSdqS ∫=

)()( qSqI ∝

光散乱の性質

• 散乱のパワースペクトルは動的構造因子のフーリエ変換に比例する。– 動的構造因子を実験的に決定可能

• 散乱光強度は入射周波数の４乗に比例– 青い空、夕焼け

• 偏光– 空の偏光

• 密度ゆらぎの増大によって大きく散乱される。– 臨界タンパク光

• エネルギー保存

),()(8

),(2

25

4

220 ω

δρδεω

πω qSnn

cRVIqI

TiS

iS ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

散乱の角度依存性について

• 散乱角),()(

8),(

22

5

4

220 ω

δρδεω

πω qSnn

cRVIqI

TiS

iS ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

ikSk

θq

θ

θ2

2|2

1

cos1

cos1

+∝

∝

∝

I

II

密度空間相関関数

• レナードジョーンズポテンシャル

• Hard core model

Critical Opalescece (臨界タンパク光)をどう理解するか？2)0()()0()()( ρρρδρδρ −== rrrG

r

r r

r

)(rG

)(rG

)(rV

)(rV

r0r0

r0

r0

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

60

120

0 2)( rr

rrVrV

密度空間相関関数１等温圧縮率との関係

2)0()()0()()( ρρρδρδρ −== rrrG )( 0)( ∞→→ rrG粒子の全数に対する揺らぎ

VTVT

NNN

NN

NNN

NN

kTPVkTLkT

NEpNrddhN

L

NEpNrddhN

L

NNNN

,2

22

,2

22

2

03

1

2

03

1

222

)/()()(ln)(

)exp(!1

)exp(!1

)(

⎭⎬⎫

⎩⎨⎧

∂∂=

⎭⎬⎫

⎩⎨⎧

∂∂=

⎭⎬⎫

⎩⎨⎧ +−−

+−=

−=−

∫∑

∫∑∞

=

−

∞

=

−

µµ

βµβ

βµβ

1963) (Huang, lnkTPVL =

∫= )(rdrN ρ

密度空間相関関数２等温圧縮率との関係

NTVT

VTVT

VV

NkTVNkTV

PkTVkTPVkTNN

,,

,2

2

,2

222

)/(

)/()()(

⎭⎬⎫

⎩⎨⎧

∂∂−=

⎭⎬⎫

⎩⎨⎧

∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂=

⎭⎬⎫

⎩⎨⎧

∂∂=−

µµ

µµ

)( ,

NFnVNP

VT

µµ

===⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xyzx yz

zw

yw

yw

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

NTNTNTT

VNP

VVP

FV ,,,

2

2 111⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂−=⎟

⎠⎞

⎜⎝⎛

∂∂−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂−=

µκ

TkTNNN κρ=−2)(

密度空間相関関数３

• 理想気体の等温圧縮率 をつかうと

• 一方、

等温圧縮率との関係

kTT ρκ 10 =

0

2)(

T

T

NNN

κκ=

−

∫= )(rdrN ρ

)()'()(')( 2 rGdrVrrdrdrNN ∫∫∫ ==− δρδρ

∫−= )(10 rdrGT

T ρκκ

密度空間相関関数４

一方、臨界点近傍では、

１．圧縮率の増大

２．密度揺らぎの増大

２．密度空間相関関数のおよぶ範囲（相関長）の増大

等温圧縮率との関係

∫−= )(10 rdrGT

T ρκκ

γ

κκ −−≈ )(0 C

T

T TT

)()( qSqI ∝

∫∫ −− == Viqr

V

iqr rGredrredqS )()0()()( 33 δρδρ

相関長の逆数がqに近づいたとき散乱強度が増大。

Critical Opalescence