A matrix approach to grazing-incidence X-ray diffraction in multilayers

Electron-electron cusp condition and asymptotic behavior for the Pauli potential in pair density functional theory

“…The Pauli energy of the pair density functional theory was introduced by a density functional analogy: the difference of the kinetic energy and a Weizsäcker-type expression 9 ͑constructed from the pair density instead of the density͒. The Pauli potential is the functional derivative of the Pauli energy with respect to the pair density.…”

High-resolution grazing-incidence x-ray diffraction for characterization of defects in crystal surface layers

“…2 In general, the sample rocking curve P GID (␦) and the position of the sample peak can be calculated numerically with the matrix method. 31,[39][40][41][42] For a periodic multilayer, there are many sample peaks at ⌽ h coinciding with different-order Bragg angles for the multilayer.…”

“…The difference is that in one scheme ͓Fig. 1͑a͔͒ the incident beam is not collimated in the Bragg plane and the diffracted waves are analyzed depending on their takeoff angle to the surface, 2,5,10,[28][29][30][31] while in the other one ͓Fig. 1͑b͔͒ the incident beam is collimated in the Bragg plane and the dependence of diffracted beam intensity on deviation of the sample from the Bragg position 1,3,4,11,12 is studied.…”

X-ray reflectivity and glancing-incidence diffraction from thin metallic Cr layers

“…This e ect has been widely used to study the oxidation kinetics of , for example, Si wafers (Awaji et al 1997, Takahashi et al 1998) and metals , Kellerman et al 1997, Stierle and Zabel 1997a . Indeed, the critical angle increases with increasing layer density, as shown in ® gure 1 (b), where the Fe layer has been substituted by a more dense W layer with the same thickness.…”

“…After the work of Sinha et al several other studies on the development of models and algorithms for calculating the scattering of X-rays at small angles have been published (Andre  and Pardo 1988, Stearns 1989, Bridou and Pardo 1994, Plotz and Lischka 1994, Sinha 1994, Stepanov et al 1995, Lu et al 1996, De Caro et al 1997, De Caro and Tapfer 1997, Voorma et al 1997 including non-specular re¯ectivity. Also the experimental aspects of XRR have been the subject of investigations aimed to develop XRR instrumentation for use at synchrotron radiation sources or combined with ultrahigh-vacuu m chambers (Niggemeier et al 1997, Lee andTseng 1998) .…”