International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 237-238

Section 4.2.5.2.3. Quasi-Bragg reflectors

D. C. Creaghb

4.2.5.2.3. Quasi-Bragg reflectors

| top | pdf |

For one interface, the reflectivity (R) and the transmissivity (T) of the surface are determined by the Fresnel equations, viz: [ R=\left | (\theta _{1}-\theta _{2})/(\theta _{1}+\theta _{2})\right | ^{2}, \eqno (4.2.5.3)]and [ T=\left | 2\theta _{1}/(\theta _{1}+\theta_{2})\right | ^{2}, \eqno (4.2.5.4)]where [\theta _{1}] and [\theta _{2}] are the angles between the incident ray and the surface plane and the reflected ray and the surface plane, respectively.

If a succession of interfaces exists, the possibility of interference between successively reflected rays exists. Param­eters that define the position of the interference maxima, the line breadths of those maxima, and the line intensity depend inter alia on the regularity in layer thickness, the interface surface roughness, and the existence of surface tilts between successive interfaces. Algorithms for solving this type of problem are incorporated in software currently available from a number of commercial sources (Bede Scientific, Siemens, and Philips). The reflectivity profile of a system having a periodic layer structure is shown in Fig. 4.2.5.3[link] . This is the reflectivity profile for a multiple-quantum-well structure of alternating aluminium gallium arsenide and indium gallium arsenide layers (Holt, Brown, Creagh & Leon, 1997[link]). Note the interference maxima that are superimposed on the Fresnel reflectivity curve. From the full width at half-maximum of these interference lines, it can be inferred that the energy discrimination of the system, ΔE/E, is 2%. The energy range that can be reflected by such a multilayer system depends on the interlayer thickness: the higher the photon energy, the thinner the layer thickness.

[Figure 4.2.5.3]

Figure 4.2.5.3| top | pdf |

The reflectivity of a multiple-quantum-well device is shown. This consists of 40 alternating layers of AlGaAs and InGaAs. Shown also, but shifted downwards on the vertical scale for the purpose of clarity, is the theoretical prediction based on standard electromagnetic theory.

Commercially available multilayer mirrors exist, and hitherto they have been used as monochromators in the soft X-ray region in X-ray fluorescence spectrometers. These monochromators are typically made of alternating layers of tungsten and carbon, to maximize the difference in scattering-length density at the interfaces. Whilst the energy resolution of such systems is not especially good, these monochromators have a good angle of acceptance for the incident beam, and reasonably high photon fluxes can be achieved using conventional laboratory sources.

A recent development of this, the Goebel mirror system, is supplied as an accessory to a commercially available diffractometer (Siemens, 1996a[link],b[link],c[link]; OSMIC, 1996[link]). This system combines the focusing capacity of a curved mirror with the energy selectivity of the multilayer system. The spacing between layers in this class of mirror multilayers can be laterally graded to enhance the incident acceptance angle. These multilayers can be fixed to mirrors of any figure to a precision of 0.3′ and can therefore can be used to form parallel beams (parabolic optical elements) as well as focused beams (elliptical optical elements) of high quality.

References

First citation Holt, S. A., Brown, A. S., Creagh, D. C. & Leon, R. (1997). The application of grazing incidence X-ray diffraction and specular reflectivity to the structural investigation of multiple quantum well and quantum dot semiconductor devices. J. Synchrotron Rad. 4, 169–174.Google Scholar
First citation OSMIC (1996). Catalogue. A new family of collimating and focusing optics for X-ray analysis. OSMIC, Michigan, USA.Google Scholar
First citation Siemens (1996a). Parallel beam optics for measurements of samples with irregularly shaped surfaces. Laboratory Report X-ray Analysis, DXRD, 13. Karlsruhe: Siemens.Google Scholar
First citation Siemens (1996b). Goebel mirrors for X-ray reflectometry investigations. Laboratory Report X-ray Analysis, DXRD, 14. Karlsruhe: Siemens.Google Scholar
First citation Siemens (1996c) Grazing incidence diffraction with Goebel mirrors. Laboratory Report X-ray Analysis, DXRD, 15. Karlsruhe: Siemens.Google Scholar








































to end of page
to top of page