Thermal properties

When installed in the white beam of an undulator, for instance, beryllium lenses must be water cooled. On the one hand, diamond has superior thermal stability and conductivity compared to beryllium, but on the other hand it has a larger attenuation coefficient.

Let us compare the heating of beryllium and diamond in the x-ray beam. For a stationary beam the relevant parameter is the ratio κ / μ. Here, κ is the thermal conductivity and μ is the attenuation coefficient. From this point of view, diamond is superior to beryllium by a factor 1 to 2 for energies between 3 and 40 keV. For a single short x-ray pulse the relevant parameter is the ratio μ / ρc, where ρc is the volumetric heat capacity. Now, from this point of view, beryllium is superior to diamond by a factor 5 to 15 for energies between 3 and 40 keV. Things are more complicated for a repetition of short pulses since then we have two relevant parameters, κ / ρc and μ / ρc.

Optical properties

From the optical point of view, the larger attenuation coefficient of diamond is partially compensated by its larger refractive decrement. At first sight, it is not evident which effect turns out to be dominant for x-ray optics at various photon energies.

In the following we compare effective aperture Deff, gain, and diffraction limited focal spot size for CRLs consisting of ideal, biconcave, parabolic 2D lenses made of beryllium and of diamond, respectively, at different photon energies. For both materials we assume a radius of curvature of R = 0.05 mm, a lens thickness D of 1 mm, a frame thickness F of 2 mm, and a web thickness d of 30 μm. (For an explanation of the lens parameters please refer to the section “Types of refractive x-ray lenses” on this page.) In each case the number of lenses is chosen such that the focal length is as close to 500 mm as possible. The gain is computed for a source with a width and height (FWHM) of 150 μm and a distance of 40 m from the CRL.

 BerylliumDiamond
Photon energy
(keV)
Focal length
(mm)
Effective aperture
(μm)
Gain
(x103)
Diffr. lim. focal
spot size (nm)
Focal length
(mm)
Effective aperture
(μm)
Gain
(x103)
Diffr. lim. focal
spot size (nm)
10494.936124.9 127492.62558.2180
20502.834119.569499.728811.481
30500.329411.853497.52638.759
40500.12567.245500.12305.851

Caution: We do not guarantee that the computed values are exactly reflecting experimental reality. The computations are based on analytical expressions assuming ideal parabolic lenses; please refer to Lengeler et al. (1999): Imaging by parabolic refractive lenses in the hard x-ray range and Schroer (2003): Hard x-ray microscopy and microanalysis with refractive x-ray lenses (for the thick lens formulae). The mass attenuation coefficients are taken from NIST.

Note that at all energies effective aperture, gain, and diffraction limited focal spot size are more favorable for the beryllium CRL than for the diamond CRL. Further computations can be found in F. Seiboth (2016): Refractive hard x-ray nanofocusing at storage ring and x-ray free-electron laser sources. There it is shown that the maximum gain achievable with beryllium CRLs is always (much) larger than for diamond CRLs, while, above approximately 17 keV, the theoretically achievable minimal diffraction limited focal spot size is slightly smaller for diamond CRLs; however, the difference is of the order of a few nm.