Chukhovskii F N, Roshchin B S
Shubnikov Institute of Crystallography of Federal Scientific Research Centre "Crystallography and Photonics", Russian Academy of Sciences, Leninsky pr. 59, Moscow, Russia, 119333.
Sci Rep. 2020 Jul 14;10(1):11547. doi: 10.1038/s41598-020-68326-2.
To describe the 1D and 2D patterns of the grazing-incidence small-angle X-ray scattering (GISAXS) from a rough fractal surface, the novel integral equations for the amplitudes of reflected and transmitted waves are derived. To be specific, the analytical expression for the 2D total intensity distribution [Formula: see text] is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular [Formula: see text] and diffuse [Formula: see text] patterns, respectively. Hereafter, [Formula: see text] is the scattering meridian angle, [Formula: see text] is the scattering azimuth angle; [Formula: see text] is the angle of incidence. By using the above analytical expressions, the 1D and 2D GISAXS patterns are numerically calculated. Some new experimental measurements of the specular reflectivity curves R([Formula: see text]) related to the fused quartz and crystal Si(111) samples have been carried out. Based on the theoretical approach developed, a direct least-squared procedure in a χ-fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.
为了描述来自粗糙分形表面的掠入射小角X射线散射（GISAXS）的一维和二维图案，推导了反射波和透射波振幅的新型积分方程。具体而言，得到了二维总强度分布[公式：见正文]的解析表达式。后者本身表示分别与GISAXS镜面[公式：见正文]和漫射[公式：见正文]图案相关的项的叠加。此后，[公式：见正文]是散射子午角，[公式：见正文]是散射方位角；[公式：见正文]是入射角。通过使用上述解析表达式，对一维和二维GISAXS图案进行了数值计算。已经对与熔融石英和晶体Si(111)样品相关的镜面反射率曲线R([公式：见正文])进行了一些新的实验测量。基于所开发的理论方法，采用χ拟合方式的直接最小二乘法从镜面GISAXS反射率数据中确定均方根粗糙度σ的相应值。
