Mathematical Sciences: Inverse Problems in Neutron Reflectometry

本研究的目的是对中子反射物理中出现的逆散射问题进行解析和数值研究。人们试图恢复中子相互作用势，它是定义在实线上的有界实值函数，在原点左侧为零，并随着空间变量趋于正无穷大而接近非零常数。用于重建的信息是对应于从左侧进入的波的反射系数的幅度，在通常的量子力学意义上定义。由于反射系数一般是复值的，位相信息的缺失是最大的难点，而实际上这一问题与光学中著名的位相恢复问题密切相关。然而，即使已知全复反射振幅，与薛定谔方程的标准逆散射公式相比，这个问题仍然呈现出一些新的方面，因为势不会衰减到零。调查员还将考虑部分已知相位数据的一些情况。在中子散射的应用中，人们通常可以假设势不存在任何束缚态，而这一假设对于主线是必不可少的。然而，当束缚态存在时，也有一些有趣的情况，因此将研究一种允许这种可能性的替代程序。利用中子反射率数据对层状材料的化学性质和磁性进行详细分析是一种新兴的技术。例如，在分子水平上研究聚合物共混物的表面和界面结构有许多应用。这反过来又是开发新的高性能材料的宝贵工具。在一个典型的实验装置中，一束中子从真空区入射到反射材料上，反射材料的材料性质只在垂直于材料表面的方向上变化。一种名为中子反射计的仪器以高精度测量反射的中子强度，它是粒子在垂直于表面的方向上动量的函数。因此，这些数据在某种程度上依赖于所谓的中子相互作用势，这是材料内部位置的函数。如果可以确定这个势，那么就可以直接推断该材料的其他物理性质。拟议的研究直接解决了解释反射率数据以确定相互作用势的问题。具体地说，调查者试图了解问题的数学结构，并开发准确可靠的计算程序。

The goal of this research is to investigate analytically and numerically an inverse scattering problem that arises in neutron reflection physics. One seeks to recover the neutron interaction potential, which is a bounded, real valued function defined on the real line, that is zero to the left of the origin, and approaches a nonzero constant as the space variable tends to positive infinity. The information to be used for the reconstruction is the amplitude of the reflection coefficient corresponding to a wave incoming from the left, defined in the usual quantum mechanical sense. Because the reflection coefficient is in general complex valued, the missing phase information is the central difficulty, and in fact this problem is closely related to the well known phase retrieval problem in optics. However, even if the full complex reflection amplitude were known, the problem still presents some new aspects, in comparison with standard formulations of inverse scattering for the Schrodinger equation, because of the fact that the potential does not decay to zero. The investigator will also consider some cases in which the phase data are partially known. In the neutron scattering application, one may usually assume that the potential does not have any bound states, and this hypothesis is essential for the main line of approach. Nevertheless the case when bound states are present is of some interest also, so an alternative procedure allowing for this possibility will be investigated. A newly emerging technology for the detailed analysis of chemical and magnetic properties of layered materials is the use of neutron reflectivity data. For example, there are many applications to the study of surface and interface structure at the molecular level for polymer blends. This in turn is a valuable tool in the development of new high performance materials. In a typical experimental set up, a beam of neutrons is incident from a vacuum region onto a reflecting material, whose material properties vary only in the direction perpendicular to the material surface. An instrument called the neutron reflectometer measures the reflected neutron intensity with high precision as a function of the particle's momentum in the direction perpendicular to the surface. These data are thus dependent in some manner on the so-called neutron interaction potential, which is a function of location within the material. If this potential can be determined, then other physical properties of the material can be directly inferred. The proposed research addresses directly the problem of interpreting reflectivity data in order to determine the interaction potential. Specifically, the investigator seeks to understand the mathematical structure of the problem, and to develop accurate and reliable computational procedures.