Scattering . 1 Basics

1.1 Recommended Books:

1] B.Chu Laser Light Scattering 2nd Ed. Academic Press 1991

2] R. J. Hunter Foundations of Colloid Science 2^nd Ed Oxford 2001

3] H. R.Cruyt Colloid Science Vol. I Elsevier 1952

4] M. Kerker The scattering of light and other electromagnetic radiation Academic Press 1969.

5] R. Richards Scattering Methods in Polymer Science Ellis Horwood 1995

Classical papers

6] Rayleigh L. Nature 3 234 [1871] 7] Debye P., Ann. Physik. 30, 57 [1909]

8] Mie G., Ann. Physik. [Leipzig] 25, 377 [1908]

9] A, Guinier X-ray diffraction In Crystals (1963) etc. Dover [1994]

1.2 Definitons:

Light is absorbed, transmitted

or scattered: d diffraction

<< d refraction (d particle diameter)

I scattered intensity at an angle q ;

I_o incident; I_T transmitted t = I_T / I₀ [1.1}

Beer Lambert Law: [1.2]

c/concentration , e/extinction coefficient.

1.3

Elastic scattering

Only changes in momentum transfer vector : energy[l] is fixed: n=1

Also c = n l v = f l [1.4} Inelastic quasielastic

1.4 Radiation types:

Sound v = 330 m s ^-1 f = 1Hz – 30 kHz l = 0.1m - 330m

Light c = 3.0 x 10 8 m s ^-1 n = 7 x1014 - 4 x 1014 Hz l = 434 - 768 nm

Neutrons v = 4000 - 400 m s^-1 m_N = 1.675E-27 kg : l = 0.1 - 1 nm

X-rays c = 3.0 x 10⁸ m s ^–1 n = 3 x10¹⁸ - 3 x 10¹⁷ Hz l = 0.1 - 1 nm

1.5 Bragg’s Law

[1.5] : Interference when phase difference is an integer number of wavelengths as in figure below.

1.6 Rayleigh Scattering

The conditions for Raleigh scattering (1871) from individual particles diameter d is that d < l/20 and refractive index, n » 1

(a) Polarisation of light.

I_o = I_Horizontal + I_Vertical. [1.6]

Unpolarised light travels along x (see figure). After scattering the light is polarised as shown in the figure along the y and z axis as there is no longitudinal component of an electromagnetic wave (i.e. in the x propagation direction). The transmitted beam along x is unpolarised. Scattering depends on the polarisability a : m = aE [1.7] m dipole moment, E Electric field strength. a is large if the HOMO-LUMO energy gap is small.

Polarisation from an unpolarised light source and viewed at different angles

(b) Dimensional analysis:

Scattering of light is due to electrons so I ~ f(a). It also must depend on the wavelength of the light, l and the distance of the observer from the scattering body and the relative refractive indices n of the particle (p) and the medium (o)

When light falls on a particle it induces a dipole moment which is proportional to the polarisability, a. The scattering field, , is then proportional to a. .Classically a is described by the Lorentz-Lorentz equation when there is no permanent dipole:

a is the radius, n=n_p/n_o and e_o is the vacuum permittivity.

So is proportional to volume a³, so the Intensity which is proportional to

E² µ a² µ a⁶ . The Inverse square law shows us that I µ r^{- -2}

So combining these gives

is dimensionless as are the values of n so this implies that

Hence .

So the intensity of the Rayleigh scatterer depends on frequency n⁴ and this explains many astronomical observations- why the sky is blue , the green flash etc.

(c) Angular dependence.

The polarisation as we have seen in angle dependent and at 90⁰ to I_o the light is completely plan polarised. The intensity must also be positive and a function which gives this is (1 + cos²(q)) where ‘1’ is the vertical component and cos²(q) is the horizontal component. So combining this with [1.12] and [1.9] gives the full Rayleigh expression:

where V_p is the particle volume and N_p is the number of particles per unit volume

NB Distance ~ 1/r²

Frequency ~ 1/l⁴ ~ n ⁴

Size ~ V_p² ~ a⁶

Concentration ~ N_p

Refractive index ~ D(n²)²

difference