FUNDAMENTALS OF X-RAY FLUORESCENCE - Total-Reflection X-Ray Fluorescence Analysis and Related Methods - page 54

where υ 1 and υ 2 are the phase velocities of the beam in media 1 and 2,
respectively. Phase velocity is the velocity at which the planes of
constant phase, for example, crests or troughs, propagate within a
medium. It is dependent on the wavelength, λ , and the medium itself.
Invacuo , the phase velocity takes the value c 0 (the light velocity)
independent of λ .

Division of Equation 1.51 by c 0 results in

n 1 cos α 1 n 2 cos α 2 (1.52)

where n 1 and n 2 are the absolute refractive indices of media 1 and 2,
respectively, which are defined by

c 0 = υ 1 ; 2 n 1 ; 2 (1.53)

The refractive index is the fraction by which the phase velocity of the
radiation is changed with respect to its vacuum value. For vacuum, the
refractive index is 1.
The wavelength of a wave is changed by refraction and the photon energy is
changed inversely to phase velocity and wavelength, though only a little bit.
Division of Equation 1.51 by the frequency leads to
λ 2 cos α 1 λ 1 cos α 2 (1.54)

Two cases can be distinguished, as demonstrated in Figure 1.29. If n 2 > n 1 , that
is, if medium 2 is optically denser than medium 1, the refracted beam in medium
2 will be deflected off the boundary. If n 2 < n 1 , that is, if medium 2 is optically
thinner than medium 1, the refracted beam in medium 2 will be deflected
toward the boundary.
The refractive index is the decisive quantity and can be derived from the so-
called Lorentz theory assuming that the quasi-elastically bound electrons of the
atoms are forced to oscillations by the primary radiation. As a result, the
oscillating electrons radiate with a phase difference. By superposition of both
radiations the primary one is altered in phase velocity. This alteration becomes
apparent by a modified refractive index, deviating from the vacuum value
n vac = 1 by a small quantity δ .
If absorption cannot be neglected but has to be taken into account, the
refractive index n has to be written as a complex quantity. Conventionally, n is
defined by

n 1 δ i β (1.55)

1. 3

where i is the imaginary unit or the square root of

3 The refractive index n can be also defined by the conjugate complex quantity 1

δ + i β .