Table of Contents
Wavelength Dependence
It is important to know that the angular range of a diffraction pattern (2θ) depends on the wavelength λ of the X-rays used in the measurement. Measurements made using laboratory diffractometers use the characteristic emissions of a metal anode, e. g. Copper or Cobalt K-alpha (λ = 1.540596 or 1.788965, respectively). In our measurement, we use synchrotron X-rays, which have a tunable wavelength—at our typical working energy of 75.0 keV, the wavelength λ = 0.165 Å. Therefore the data you receive will have a different angular range than measurements using a laboratory diffractometer and the peaks will show up at different 2θ positions compared to your lab measurements. Therefore, the data cannot be directly compared without normalization.
To compare patterns measured at different wavelength, one can normalize the 2θ value using the respective wavelength to an absolute scale defined by the eponymous independent variable momentum transfer Q, using the equation
Q = 4π∗sin(2θ/2)/λ,
or invert the relationship to convert from Q back to 2θ by
2θ = 2∗arcsin(Q∗λ/(4π)).
This relationship comes directly from the Bragg equation
nλ=2dhkl∗sin(θ),
where momentum transfer is related to the d-spacing by
dhkl = 2∗π/Q.
One can see that while 2θ is relevant to the reference frame of the measurement geometry, Q is more directly relevant to the length-scales probed in the structure-of-interest.
Note: For the normalization the values for 2θ have to be in radians. To convert the 2θ values from degree to radians you can use this formula:
2θradians = 2θdegree∗π/180
Converting between angular ranges
In principle one can convert the angular range of a dataset collected using one wavelength to the angular range of a new wavelength by combining the first two equations with the corresponding original and final λ values. However, this is not recommended, because both the angular range and wavelength can have implications on data modeling. E.g., many artefacts are angle dependent due to the sample and detector geometries, while the wavelength influences the X-ray atomic form factors and absorption.
Thus we recommend not to do such a conversion. Good software tools should allow you to specify the wavelength or energy used to collect the pattern.
Accessible Q-range Comparison
We compare the Q-ranges accessible to measurements by us using synchrotron radiation to those expected for typical laboratory instrumentation. The conversions of 2θ to Q for different measurements are shown in the figure below. For laboratory instrumentation, we have chosen a maximum angle of 160°, although may actually be quite high for many systems.
Notably, for total scattering data, it is clear that we can collect to a much higher Qmax than laboratory systems, meaning that a higher spatial resolution can be achieved while minimizing artefacts that come from truncation of data at lower values. Additionally, our high-resolution ,measurements cover a range wider than what can be achieved with a Cu-anode system, and commensurate with measurements up to approximately 60° using a Mo anode or 45° using a Ag anode.