Reflectivity

Reflectivity refers to the measurement of the intensity of reflection off of a flat interface. The term both describes the physical phenomenon, as well as the experimental technique.

X-ray Reflectivity (XRR or XR) and neutron reflectivity (NR) are techniques which measure the intensity of reflected radiation as a function of angle (where, by definition for specular reflectivity, the incident and exit angles are equal; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \theta_i = \theta_f} ). A plot of reflectivity (R) versus angle yields the reflectivity curve. For XR and NR, the data is typically plotted as a function of the momentum transfer parallel to the film normal:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_z = \frac{4\pi}{\lambda} \sin \theta }

Reflectivity calculation for a thin film.

Off-Specular Reflectivity

TBD

Mathematical form

In its simplest form, the Fresnel reflectivity can be given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_F = \frac{k_z-\tilde{k}_z}{k_z+\tilde{k}_z} }

Where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{k}_z = - \sqrt{ n^2 k^2_0 - |k_{\parallel}|^2 } }

And n is the complex refractive index of the substrate. The idealized uncorrelated roughness can be characterized by a mean standard deviation of the height Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \sigma = \sqrt{ \left \langle h^2 \right \rangle }} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_S = R_F e^{ -2 \sigma^2 k_z \tilde{k}_z } }

For a substrate with a single continuous layer of thickness h (e.g. a uniform thin film), the reflectivity becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_S = \frac{ r_{01} + r_{12} e^{ 2i \tilde{k}^1_z h } }{ 1 + r_{01} r_{12} e^{ 2i \tilde{k}^1_z h } } }

where

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is the perpendicular component of the wave-vector (in medium j). The reflectivity coefficients are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij} = \frac{ \tilde{k}^i_z - \tilde{k}^j_z }{ \tilde{k}^i_z + \tilde{k}^j_z } }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle r_{01}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle r_{12}} for the vacuum-layer and layer-substrate interfaces, respectively. This is called the 'one-box model'.

Parratt formalism

TBD

• L. G. Parratt Surface Studies of Solids by Total Reflection of X-Rays Phys. Rev. 1954, 95, 359. doi: 10.1103/PhysRev.95.359

See Also

• Fresnel plot
• Oscillations below the critical angle
• DWBA: A formalism for modeling GISAXS data, including reflection effects.