ABSTRACT
A planar X-ray wavefield is created by the superposition of two plane X-ray waves: E0 = e0E0 exp[2pi(v0t – K0 ◊ r)] (5.1)and EH = eHEH exp[2pi(v0t – KH ◊ r)] (5.2) Here, e0 and eH are the polarization vectors and K0 and KH are the X-ray propagation vectors with |K0| = |KH| = |K| = K = l-1, (5.3)
where l is the X-ray wavelength. Since Eg = hv0 and c = lv0, the X-ray energy Eg and l are related via lEg = hc = 1.23984 keV∙nm. The two waves are coherent with identical frequencies. Thus, one can relate the two E-field vectors by amplitude and phase factors via E RE iH = 0 exp( )j (5.4)with R = IH/I0 = |EH|2/|E0|2. (5.5) Further, the propagation direction of the two plane waves is related by KH = K0 + H. (5.6) Thus, it is straightforward to show that for the total wavefield intensity (normalized to the incident intensity I0) one obtains the following expression: I R R v R R v z dH H HH r= + + - ◊ = + + -1 2 2 1 2 2cos( ) cos( / ) .p p (5.7) Here, r is the distance with respect to a chosen reference point and thus zH is the projection of this distance along H, that is, normal to the wavefield planes. The phase v (not to be confused with the frequency v0) between the two plane waves depends of course on the origin chosen for r. When using a standing wave created by single-crystal Bragg diffraction, the origin of r is the same as the origin chosen for the structure factor of the crystal which produces the XSW. Further, the wavefield spacing dH corresponds, in this case, exactly to the chosen diffraction plane spacing and H = [hkl] denotes the chosen diffraction vector with |H| = H = 1/dH,where (hkl) are the Miller indices of the diffraction planes. Equation (5.7) describes a planar wavefield, which is modulated in the direction of H with the spacing dH. The position of the maxima and minima in the direction of H is determined by the value of v. 5.3 XSW AnalysisIf the dipole approximation4 for the photoabsorption process can be applied, the intensity I of the X-ray excited photoelectrons, Auger 4See footnote 1 on page 1.
