$$\renewcommand\AA{\text{Å}}$$

# X-ray scattering potentials¶

## periodictable.xsf¶

This module has one class and nine fuctions.

Xray
X-ray scattering properties for the elements.

The following attributes are added to each element:

Xray.sftable()
Three column table of energy vs. scattering factors f1, f2.
Xray.scattering_factors()
Returns f1, f2, the X-ray scattering factors for the given wavelengths interpolated from sftable.
Xray.f0()
Returns f0 for the given vector Q, with Q[i] in $$[0, 24\pi]$$-1.
Xray.sld()
Returns scattering length density (real, imaginary) for the given wavelengths or energies.

The following functions are available for X-ray scatting information processing:

xray_wavelength()
Finds X-ray wavelength in angstroms given energy in keV.
xray_energy()
Finds X-ray energy in keV given wavelength in angstroms.
init()
Initializes a periodic table with the Lawrence Berkeley Laboratory Center for X-Ray Optics xray scattering factors.
init_spectral_lines()
Sets the K_alpha and K_beta1 wavelengths for select elements.
sld_table()
Prints the xray SLD table for the given wavelength.
xray_sld()
Computes xray scattering length densities for molecules.
index_of_refraction()
Express xray scattering length density as an index of refraction
mirror_reflectivity()
X-ray reflectivity from a mirror made of a single compound.
xray_sld_from_atoms()
The underlying scattering length density calculator. This works with a dictionary of atoms and quantities directly.
emission_table()
Prints a table of emission lines.
K_alpha, K_beta1 (Å):
X-ray emission lines for elements beyond neon, with $$K_\alpha = (2 K_{\alpha 1} + K_{\alpha 2})/3$$.
X-ray scattering factors:
Low-Energy X-ray Interaction Coefficients: Photoabsorption, scattering and reflection for E in 30 to 30,000 eV, and Z in 1 to 92.

### Emission line tables¶

Data for the $$K_\alpha$$ and $$K_\beta$$ lines comes from [#Deslattes2003], with the full tables available at http://www.nist.gov/pml/data/xraytrans/index.cfm. Experimental Values are used, truncated to 4 digits of precision to correspond to the values for the subset of elements previously defined in the periodictable package.

### X-ray f1 and f2 tables¶

The data for the tables is stored in the periodictable/xsf. directory. The following information is from periodictable/xsf/read.me, with minor formatting changes:

These [*.nff] files were used to generate the tables published in reference [1]. The files contain three columns of data:

Energy(eV), f_1, f_2,

where f_1 and f_2 are the atomic (forward) scattering factors. There are 500+ points on a uniform logarithmic mesh with points added 0.1 eV above and below “sharp” absorption edges. The tabulated values of f_1 contain a relativistic, energy independent, correction given by:

$Z^* = Z - (Z/82.5)^{2.37}$

Note

Below 29 eV f_1 is set equal to -9999.

The atomic photoabsorption cross section, $$\mu_a$$, may be readily obtained from the values of $$f_2$$ using the relation:

$\mu_a = 2 r_e \lambda f_2$

where $$r_e$$ is the classical electron radius, and $$\lambda$$ is the wavelength. The index of refraction for a material with N atoms per unit volume is calculated by:

$n = 1 - N r_e \lambda^2 (f_1 + i f_2)/(2 \pi).$

These (semi-empirical) atomic scattering factors are based upon photoabsorption measurements of elements in their elemental state. The basic assumption is that condensed matter may be modeled as a collection of non-interacting atoms. This assumption is in general a good one for energies sufficiently far from absorption thresholds. In the threshold regions, the specific chemical state is important and direct experimental measurements must be made.

These tables are based on a compilation of the available experimental measurements and theoretical calculations. For many elements there is little or no published data and in such cases it was necessary to rely on theoretical calculations and interpolations across Z. In order to improve the accuracy in the future considerably more experimental measurements are needed.

Note that the following elements have been updated since the publication of Ref. [1] in July 1993. Check http://henke.lbl.gov/optical_constants/update.html for more recent updates.

Element Updated Energy Range (eV)
Mg Jan 2011 10-1300
Zr Apr 2010 20-1000
La Jun 2007 14-440
Gd Jun 2007 12-450
Sc Apr 2006 50-1300
Ti Aug 2004 20-150
Ru Aug 2004 40-1300
W Aug 2004 35-250
Mo Aug 2004 25-60
Be Aug 2004 40-250
Mo Nov 1997 10-930
Fe Oct 1995 600-800
Si Jun 1995 30-500
Au Jul 1994 2000-6500
Mg,Al,Si Jan 1994 30-200
Li Nov 1994 2000-30000
Data available at:
 [1] (1, 2) B. L. Henke, E. M. Gullikson, and J. C. Davis. “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92”, Atomic Data and Nuclear Data Tables 54 no.2, 181-342 (July 1993).
 [2] R. D. Deslattes, E. G. Kessler, Jr., P. Indelicato, L. de Billy, E. Lindroth, and J. Anton. Rev. Mod. Phys. 75, 35-99 (2003).
class periodictable.xsf.Xray(element)

Bases: object

X-ray scattering properties for the elements. Refer help(periodictable.xsf) from command prompt for details.

f0(Q)

Isotropic X-ray scattering factors f0 for the input Q.

Parameters: Q : float or vector in $$[0, 24\pi]$$ | Å-1 X-ray scattering properties for the elements. f0 : float Values outside the valid range return NaN.

Note

f0 is often given as a function of $$\sin(\theta)/\lambda$$ whereas we are using $$Q = 4 \pi \sin(\theta)/\lambda$$, or in terms of energy $$Q = 4 \pi \sin(\theta) E/(h c)$$.

Reference:
D. Wassmaier, A. Kerfel, Acta Crystallogr. A51 (1995) 416. http://dx.doi.org/10.1107/S0108767394013292
scattering_factors(energy=None, wavelength=None)

X-ray scattering factors f’, f’’.

Parameters: energy : float or vector | keV X-ray energy. scattering_factors : (float, float) Values outside the range return NaN.

Values are found from linear interpolation within the Henke Xray scattering factors database at the Lawrence Berkeley Laboratory Center for X-ray Optics.

sld(wavelength=None, energy=None)

X ray scattering length density.

Parameters: wavelength : float or vector | Å Wavelength of the X-ray. energy : float or vector | keV Energy of the X-ray (if wavelength not specified). sld : (float, float) | Å-2 (real, imaginary) X-ray scattering length density. TypeError : neither wavelength nor energy was specified.

The scattering length density is $$r_e N (f_1 + i f_2)$$. where $$r_e$$ is the electron radius and $$N$$ is the number density. The number density is $$N = \rho_m/m N_A$$, with mass density $$\rho_m$$ molar mass $$m$$ and Avogadro’s number $$N_A$$.

The constants are available directly:

$$r_e$$ = periodictable.xsf.electron_radius

$$N_A$$ = periodictable.constants.avogadro_number

Data comes from the Henke Xray scattering factors database at the Lawrence Berkeley Laboratory Center for X-ray Optics.

scattering_factors_units = ['', '']
sftable

X-ray scattering factor table (E,f1,f2)

sftable_units = ['eV', '', '']
sld_units = ['1e-6/Ang^2', '1e-6/Ang^2']
periodictable.xsf.init(table, reload=False)
periodictable.xsf.init_spectral_lines(table)

Sets the K_alpha and K_beta1 wavelengths for select elements

periodictable.xsf.xray_energy(wavelength)

Convert X-ray wavelength to energy.

Parameters: wavelength : float or vector | Å energy : float or vector | keV

Wavelength can be converted to energy using

$E = h c / \lambda$

where:

$$h$$ = planck’s constant in eV·s

$$c$$ = speed of light in m/s

periodictable.xsf.xray_wavelength(energy)

Convert X-ray energy to wavelength.

Parameters: energy : float or vector | keV wavelength : float | Å

Energy can be converted to wavelength using

$\lambda = h c / E$

where:

$$h$$ = planck’s constant in eV·s

$$c$$ = speed of light in m/s

periodictable.xsf.xray_sld(compound, density=None, natural_density=None, wavelength=None, energy=None)

Compute xray scattering length densities for molecules.

Parameters: compound : Formula initializer Chemical formula. density : float | g·cm-3 Mass density of the compound, or None for default. natural_density : float | g·cm-3 Mass density of the compound at naturally occurring isotope abundance. wavelength : float or vector | Å Wavelength of the X-ray. energy : float or vector | keV Energy of the X-ray, if wavelength is not specified. sld : (float, float) | 10-6Å-2 (real, imaginary) scattering length density. AssertionError : density or wavelength/energy is missing.
periodictable.xsf.xray_sld_from_atoms(*args, **kw)

Deprecated since version 0.91: xray_sld() now accepts a dictionary of {atom: count} directly.

periodictable.xsf.emission_table(table=None)

Prints a table of emission lines.

Parameters: table : PeriodicTable The default periodictable unless a specific table has been requested. None

Example

>>> emission_table() # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
El  Kalpha  Kbeta1
Ne 14.6102 14.4522
Na 11.9103 11.5752
Mg  9.8902  9.5211
Al  8.3402  7.9601
Si  7.1263  6.7531
...

periodictable.xsf.sld_table(wavelength=None, table=None)

Prints the xray SLD table for the given wavelength.

Parameters: wavelength = Cu K-alpha : float | Å X-ray wavelength. table : PeriodicTable The default periodictable unless a specific table has been requested. None

Example

>>> sld_table() # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
X-ray scattering length density for 1.5418 Ang
El    rho   irho
H   1.19   0.00
He   1.03   0.00
Li   3.92   0.00
Be  13.93   0.01
B  18.40   0.01
C  18.71   0.03
N   6.88   0.02
O   9.74   0.04
F  12.16   0.07
Ne  10.26   0.09
Na   7.98   0.09
Mg  14.78   0.22
...

periodictable.xsf.plot_xsf(el)

Plots the xray scattering factors for the given element.

Parameters: el : Element None
periodictable.xsf.index_of_refraction(compound, density=None, natural_density=None, energy=None, wavelength=None)

Calculates the index of refraction for a given compound

Parameters: compound : Formula initializer Chemical formula. density : float | g·cm-3 Mass density of the compound, or None for default. natural_density : float | g·cm-3 Mass density of the compound at naturally occurring isotope abundance. wavelength : float or vector | Å Wavelength of the X-ray. energy : float or vector | keV Energy of the X-ray, if wavelength is not specified. n : float or vector | unitless index of refraction of the material at the given energy

Formula taken from http://xdb.lbl.gov (section 1.7) and checked against http://henke.lbl.gov/optical_constants/getdb2.html

periodictable.xsf.mirror_reflectivity(compound, density=None, natural_density=None, energy=None, wavelength=None, angle=None, roughness=0)

Calculates reflectivity of a thick mirror as function of energy and angle

Parameters: compound : Formula initializer Chemical formula. density : float | g·cm-3 Mass density of the compound, or None for default. natural_density : float | g·cm-3 Mass density of the compound at naturally occurring isotope abundance. roughness : float | Å High-spatial-frequency surface roughness. wavelength : float or vector | Å Wavelength of the X-ray. energy : float or vector | keV Energy of the X-ray, if wavelength is not specified. angle : vector | ° Incident beam angles. reflectivity : matrix matrix of reflectivity as function of (angle, energy)

Formula taken from http://xdb.lbl.gov (section 4.2) and checked against http://henke.lbl.gov/optical_constants/mirror2.html