X-ray Reflectivity

Detlef Smilgies


Fresnel formulae

The complete theory of reflection and refraction of an electromagnetic wave at an interface is given by the Fresnel formulae (around 1830). It took 30 years after the discovery of x-rays, until first Kiessig in the 1930's and later Abeles and Parratt in the 1950's applied the Fresnel theory to x-rays, and thus created a tool to study the structure at interfaces on a scale ranging from 10 to several 1000 : X-ray Reflectivity.

From the boundary conditions we already know the wave vector on either side of the interface. Thus we can write down the boundary condition for the tangential electric fields, where we include the continous part of the wave  exp{i (k|| r - ωt)}  across the interface in E||:
E|| exp(i kz rz) + r E|| exp(-i kz rz)  = t E|| exp(i kz' rz)
The reflection factor r and the transmission factor t are complex, i.e. carry information about both the phase and the magnitude of the reflected and the transmitted (or refracted) wave as compared to the incident wave. Choosing the interface boundary at rz=0, this condition simplifies to
1 + r = t
A second boundary condition is obtained for the parallel component of the magnetic field H. The magnetic field is related to E via the Maxwell equation
-∂/∂t (μ μ0 H) = ∇ ⨯ E
yielding for a plane electromagnetic wave: 
ω μ μ0 H = kE
The second boundary condition can thus be rewritten in terms of the electric field  
(k E)|| /μ  =  (k' ⨯ E')||/μ'
and reads for the case of E parallel to the interface, and considering that the z-components of the wave vectors for the reflected and transmitted waves are given by -kz and k'z, respectively :
( kz - r kz ) / μ = t k'z / μ'
With μ = μ' = 1 for non-magnetic materials, we obtain the Fresnel formulae
r = (kz - kz') / (kz + kz')
t = 2 kz / (kz + kz')
The reflected and transmitted intensities are proportional to || E ||2 and thus
RF = | r |2
TF = Re(k'z/kz) | t |2

The q-4 law

RF and | t |2 are plotted below. Above the critical angle RF falls off like kz-4 which can be seen by expanding kz' and r :

n2-1 ≈ -2δ = -αc2
kz'/kz = sqrt( (n2-1) k2/kz2 + 1) ≈ sqrt (- αc2 k2/kz2 + 1) ≈ 1 - kzc2/ 2 kz2
r   = (1-kz'/kz) / (1+kz'/kz) ≈ kzc2 / 4 kz2
for kz > kzc = k αc where kzc is the z-component of k at the critical angle.

Finally we introduce the scattering vector q as q = kf - ki., where ki and kf are the wave vectors of the incident and scattered wave, respectively. In specular reflectivity only the z-component of q is non-vanishing and is easily obtained from k=ki

qz = 2 kz
By substituting kz by qz we derive the famous q-4 law:
RF  =  kzc4 / qz4

Relative units

A frequently used way of presenting reflectivity data is relative units. If we introduce

x  =  kz/kz,c =  α /αc  =  q/qc
y  =  β/ δ
x' =  ( x2-1+i y )0.5
where the index c denotes the respective quantity taken at the critical angle, the Fresnel reflectivity can be written as
R = |r|2 = | (x-x') / (x+x') |2
T = |t|2 = | 2 / (x+x') |2
In the following R and T are shown for different values of  y = β/δ.

The effect of β is that the sharp singularites at the critical angle are smoothed out. If β were larger than 0.5 δ, there would hardly be a clear-cut critical angle any more. However, such exotic conditions can be only met right at an absorption edge
(for example calcium at 4039 eV).

| t |2 describes the strength of the refracted wave immediately at the interface. At the critical angle, incident and reflected wave are in phase, so that the field amplitude is doubled. Close to α=0 the fields are out of phase, i.e. cancel each other, so that and the amplitude of the transmitted wave goes to 0.

Penetration depth, escape depth, and scattering depth

Other quantities important for the experiment that can be easily extracted from our calculation above are the penetration depth Λ and the scattering depth Λs. We will give here a more complete treatment than needed for the specular reflectivity alone, as the arguments also apply for any other kind of grazing incidence scattering technique.

Λ is related to the damping of the incident wave perpendicular to the interface which is given by the imaginary part of the z-component of the transmitted wave vector

Λ = 2π / Im(kz')
Similarly, there is an escape depth associated with a photon transmited at a finite depth below the sample surface:

Λescape = 2π / Im(kf,z')

where kf' is the wave vector of this scattered or emitted photon inside the material.

The scattering depth Λs, sometimes also called the information depth, takes into account, that for any scattering signal we detect, both the path of the incident wave to the scattering event as well as the path of the scattered wave through the medium have to be considered:
1/Λs = 1/Λ  +  1/Λescape     
In the case of specular reflectivity the exit wave vector kf,z' is simply -kz' and qz' = 2 kz' and hence
Λs = 2π / Im(qz') = Λ/2
The figure shows the penetration depth as function of x = α/αc for different values of β/δ. Above the critical angle (x>1) the penetration depth is controlled mainly by x-ray absorption, while below the critical angle refraction dominates. Note that the limiting value for the penetration depth at zero incidence is about 50, independent of element.


Appendix: Practical units

In our derivation we used k to characterize the incident wave. In an experiment we would usually measure the angle of incidence α:

kz = k sin α ≈ k α (rad)
The scattering vector q is given by
q = 2 k sin α = 2 kz
With 2 k = 4π/λ = 12.566/λ and E = hν = hc /λ = 12.4 keV / λ we can use the accidental similarity of the numerical values of 4π and hc (keV ) to write
q (-1)  =  E (keV) sin α   ≈   E (keV) α (mrad)
which is very convenient for some first numerical estimates during an experiment.