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xrayAttenuation

This library provides the ability to compute transmission / absorption behavior of X-rays at different energies for all elements below Z < 93.

Mass attenuation coefficients μ_m/ρ are used in the Beer-Lambert law to compute the absorption behavior. For most energies these are first computed from scattering form factors.

The library basically provides Nim types for all different elements that are supported via a Element[Z: static int] base type and compounds of multiple elements via a Compound type, which stores the element and number of atoms of that type.

Example:

import xrayAttenuation
  
let ar = Argon.init() # generate an Argon instance
ar.plotAttenuation() # generate a plot of the attenuation factors against energy

# compute a density at known pressure and temperature (the molar mass is filled automatically
# in `init`)
let ρ_Ar = density(1050.mbar.to(Pascal), 293.K, ar.molarMass)
# use our density and a desired length to generate a plot of the transmission in 3cm Argon
ar.plotTransmission(ρ_Ar, 3.cm.to(m))
# compute an individual absorption length at a desired energy
echo absorptionLength(2.5.keV, numberDensity(ρ_Ar, ar.molarMass),
                      ar.f2.eval(2.5))

# this can also be computed for compounds. To make construction of a
# compound easier, we can use the `compound` macro
block SimpleCompound:
  let Si = Silicon.init()
  let N = Nitrogen.init()
  let Si₃N₄ = initCompound((Si, 3), (N, 4))
  Si₃N₄.plotTransmission(Si₃N₄.ρ, # for some common compounds we have a table of densities,
                                 # which are filled upon compound initialization
                         300.nm.to(Meter))
block CompoundMacro:
  # or simpler using the macro (i.e. no initialization of `Si` and `N` needed:
  let Si₃N₄ = compound (Si, 3), (N, 4)
  Si₃N₄.plotTransmission(Si₃N₄.ρ, 300.nm.to(Meter))

# finally, reflectivities can be computed
let Au = Gold.init()
# note: currently densities of elements are not read by default. 
discard Au.plotReflectivity(19.32.g•cm⁻³,
                            θ = 0.5.°) # incidence angle of 0.5° from surface
# the procedure (currently, but that will be changed) returns the DF containing the reflectivity

See the API (docs will be published soon) / source file on how to compute individual properties directly instead of only plotting the data.

Data

The data used for the calculations are a mix of the dataset by NIST:

https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients

and the Center for X-ray Optics (“Henke”):

https://henke.lbl.gov/optical_constants/

and specifically:

https://henke.lbl.gov/optical_constants/asf.html

Resources

This section lists a few resources that can be useful for anyone who wants to understand where the equations used here come from.

General

The X-ray data booklet:

https://xdb.lbl.gov/xdb-new.pdf

gives a good overview of all the topics required by this library (and more of course!)

Note though that the book is short on maths and some the equations are hard to implement correctly (grazing angle reflectivity for example).

And of course the main website of it:

https://xdb.lbl.gov/

In particular this PDF of “Useful formulas” is pretty handy:

https://xdb.lbl.gov/Section5/Sec_5-5.pdf

Further, NIST provides another database of scattering factors and other numbers: https://physics.nist.gov/PhysRefData/FFast/html/form.html

Other libraries for X-ray calculations

The DarpanX library provides similar functionality to this library, with a focus on reflectivities for single and multi-layer mirrors (but it’s implemented in Python and rather slow):

Reflectivity

The paper “Reflection of X-rays from a rough surface at extremely small grazing angles”

https://doi.org/10.1364/OE.23.024220

was very useful in getting the basic reflectivity code working initially.

The following ~book contains an introduction about reflectivity, including the derivation of the Fresnel equations (which provide the basis to compute reflectivity and transmission), as well as

https://www.afc.asso.fr/images/reflecto2018/reflectie.pdf

Very short introduction to the topic from a lab course manual on X-ray reflectometry at Uni Siegen:

https://www.hep.physik.uni-siegen.de/teaching/masterlab/manuals/XRR-2019-manual.pdf

Wikipedia on Fresnel equations:

https://en.wikipedia.org/wiki/Fresnel_equations

See also the paper about the DarpanX library, in particular the appendix for an overview of the basic approach to compute reflectivities.

Surface roughness

Mentions the origin of the dampening factor to Rayleigh & acoustic waves.

https://www.classe.cornell.edu/~dms79/refl/XR-Roughness.html

also mentions Névot–Croce factors as a generalization of that.

Paper: “Influence of surface and interface roughness on X-ray and extreme ultraviolet reflectance: A comparative numerical study”

seems to provide a good introduction.