\(\renewcommand\AA{\text{Å}}\)
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 \([0, 24\pi]\) Å-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.
Note
For custom tables, use init()
and
init_spectral_lines()
to set the data.
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
- Qfloat or vector in \([0, 24\pi]\) | Å-1
X-ray scattering properties for the elements.
- Returns
- f0float
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
- energyfloat or vector | keV
X-ray energy.
- Returns
- 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
- wavelengthfloat or vector | Å
Wavelength of the X-ray.
- energyfloat or vector | keV
Energy of the X-ray (if wavelength not specified).
- Returns
- sld(float, float) | Å-2
(real, imaginary) X-ray scattering length density.
- Raises
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
= ['', '']¶
-
property
sftable
¶ X-ray scattering factor table (E,f1,f2)
-
sftable_units
= ['eV', '', '']¶
-
sld_units
= ['1e-6/Ang^2', '1e-6/Ang^2']¶
-
-
periodictable.xsf.
emission_table
(table=None)¶ Prints a table of emission lines.
- Parameters
- tablePeriodicTable
The default periodictable unless a specific table has been requested.
- Returns
None
Example
>>> emission_table() 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.
index_of_refraction
(compound, density=None, natural_density=None, energy=None, wavelength=None)¶ Calculates the index of refraction for a given compound
- Parameters
- compoundFormula initializer
Chemical formula.
- densityfloat | g·cm-3
Mass density of the compound, or None for default.
- natural_densityfloat | g·cm-3
Mass density of the compound at naturally occurring isotope abundance.
- wavelengthfloat or vector | Å
Wavelength of the X-ray.
- energyfloat or vector | keV
Energy of the X-ray, if wavelength is not specified.
- Returns
- nfloat or vector | unitless
index of refraction of the material at the given energy
- Notes
Formula taken from http://xdb.lbl.gov (section 1.7) and checked against http://henke.lbl.gov/optical_constants/getdb2.html
-
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.
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
- compoundFormula initializer
Chemical formula.
- densityfloat | g·cm-3
Mass density of the compound, or None for default.
- natural_densityfloat | g·cm-3
Mass density of the compound at naturally occurring isotope abundance.
- roughnessfloat | Å
High-spatial-frequency surface roughness.
- wavelengthfloat or vector | Å
Wavelength of the X-ray.
- energyfloat or vector | keV
Energy of the X-ray, if wavelength is not specified.
- anglevector | °
Incident beam angles.
- Returns
- reflectivitymatrix
matrix of reflectivity as function of (angle, energy)
- Notes
Formula taken from http://xdb.lbl.gov (section 4.2) and checked against http://henke.lbl.gov/optical_constants/mirror2.html
-
periodictable.xsf.
plot_xsf
(el)¶ Plots the xray scattering factors for the given element.
- Parameters
el : Element
- Returns
None
-
periodictable.xsf.
sld_table
(wavelength=None, table=None)¶ Prints the xray SLD table for the given wavelength.
- Parameters
- wavelength = Cu K-alphafloat | Å
X-ray wavelength.
- tablePeriodicTable
The default periodictable unless a specific table has been requested.
- Returns
None
Example
>>> sld_table() 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.
xray_energy
(wavelength)¶ Convert X-ray wavelength to energy.
- Parameters
wavelength : float or vector | Å
- Returns
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_sld
(compound, density=None, natural_density=None, wavelength=None, energy=None)¶ Compute xray scattering length densities for molecules.
- Parameters
- compoundFormula initializer
Chemical formula.
- densityfloat | g·cm-3
Mass density of the compound, or None for default.
- natural_densityfloat | g·cm-3
Mass density of the compound at naturally occurring isotope abundance.
- wavelengthfloat or vector | Å
Wavelength of the X-ray.
- energyfloat or vector | keV
Energy of the X-ray, if wavelength is not specified.
- Returns
- sld(float, float) | 10-6Å-2
(real, imaginary) scattering length density.
- Raises
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.
xray_wavelength
(energy)¶ Convert X-ray energy to wavelength.
- Parameters
energy : float or vector | keV
- Returns
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