Refractive Index

Attempt of a Rigorous Derivation

Detlef Smilgies

The Maxwell equations in the presence of a medium are given in SI units (m kg s A) as:

Ñ D = 0 Ñ ´ E = -¶/¶t B
Ñ B = 0 Ñ ´ H = ¶/¶t D

Furthermore there are the phenomenological relations between the fields

D = e e₀ E B = m m₀ H

In vacuum e=m=1, in a medium a microscopic theory is necessary to determine e and m . The propagation of an electromagnetic wave through the medium is determined by the wave equation, which can be derived from the above equations by considering :

Ñ ´ (Ñ ´ E) = -Ñ ´ (¶/¶t B)

The left-hand side simplifies with the help of ÑD = 0 to

Ñ ´ (Ñ ´ E) = Ñ (Ñ E) - (Ñ²) E = -Ñ²E

The right-hand side can be rewritten as

-Ñ ´ ¶/¶t B = ¶/¶t Ñ ´ m m₀H = m m₀ ¶²/¶t²D = e e₀ m m₀ ¶²/¶t²E

yielding

Ñ² E = 1/c²¶²/¶t²E

with the speed of light in the medium given by

c = 1/sqrt( e e₀ m m₀).

The refractive index n is then defined as the ratio between the speed of light in vacuum (e=m=1) and the speed of light in the medium:

n = c₀ / c = sqrt(e m).

Solutions of the homogeneous wave equation are plane waves

E = E₀ exp(i k r - wt)

which have to fulfill the dispersion relation

w = c k = (c₀/n) k

where k = || k ||. The dispersion relation is obtained by inserting the solutions into the wave equation.

X-ray absorption

X-ray absorption is the attenuation of an x-ray beam through a layer of thickness x as given by Beer's law:

I(x) = I₀ exp(-mx)

Both the photoelectric effect and inelastic scattering (Compton scattering) contribute to this loss of intensity. If we are not interested in the detailed nature of the absorption process, we can introduce the absorption as an imaginary part b of the refractive index: If the photon beam in a medium with direction k/k is described as a damped plane wave:

E(x,t) = E0 exp{i (n kx-wt)} exp(-m (k/k) x/2)

the attenuation of the wave can formally be included in n by setting

b = m/2k

and we obtain

E(x,t) = E0 exp{i ( [1-d+ib] kx - wt)}

Note that from now on we have to be careful with boundary conditions and choosing the correct sign before ib.

A deeper justification for this approach can be given within the framework of linear response theory, the most important result of which is that d amd b are not independent of each other, but related by the Kramers-Kronig relations.

Connection of the refractive index to the microscopic scattering theory

Finally it should be mentioned that the macroscopic constants d and b are related to the electron density including the dispersive (f ') and absorptive (f '') corrections:

d = r₀ (l²/2p) (r/A) (Z-f ')
b = r₀ (l²/2p) (r/A) f "

where r and A are the mass density and atomic weight of the atom, respectively, r₀ = e²/(4pe₀ mc) = 2.818 fm is the classical electron radius, and l= hc/E the x-ray wave length. Z is the number of electrons per atom. For a chemical compound the d and b values can be obtained from the mass density, atomic mass numbers A_i and atomic charges Z_i and the dispersion corrections f_i', f_i" for the individual atoms:

d = r₀ l²/2p (r/S_i A_i) S_i(Z_i - f_i')
b = r₀ l²/2p (r/S_i A_i) S_if_i"

Sometimes scattering length densities are introduced, to compare to the formalism as used in neutron scattering. The real and imaginary part of the scattering length density b are given by:

Re(b) = 2p d /l² Im(b) = 2p b /l²

Further development of this page:

Kramers-Kronig dispersion relations