Fullagar Wilfred K, Paziresh Mahsa, Latham Shane J, Myers Glenn R, Kingston Andrew M
Applied Mathematics, RSPE, Oliphant Building 60, Mills Road, Australian National University, Canberra, ACT 2601, Australia.
Acta Crystallogr B Struct Sci Cryst Eng Mater. 2017 Aug 1;73(Pt 4):675-695. doi: 10.1107/S2052520617009222. Epub 2017 Jul 28.
In statistics, the index of dispersion (or variance-to-mean ratio) is unity (σ/〈x〉 = 1) for a Poisson-distributed process with variance σ for a variable x that manifests as unit increments. Where x is a measure of some phenomenon, the index takes on a value proportional to the quanta that constitute the phenomenon. That outcome might thus be anticipated to apply for an enormously wide variety of applied measurements of quantum phenomena. However, in a photon-energy proportional radiation detector, a set of M witnessed Poisson-distributed measurements {W, W,… W} scaled so that the ideal expectation value of the quantum is unity, is generally observed to give σ/〈W〉 < 1 because of detector losses as broadly indicated by Fano [Phys. Rev. (1947), 72, 26]. In other cases where there is spectral dispersion, σ/〈W〉 > 1. Here these situations are examined analytically, in Monte Carlo simulations, and experimentally. The efforts reveal a powerful metric of quanta broadly associated with such measurements, where the extension has been made to polychromatic and lossy situations. In doing so, the index of dispersion's variously established yet curiously overlooked role as a metric of underlying quanta is indicated. The work's X-ray aspects have very diverse utility and have begun to find applications in radiography and tomography, where the ability to extract spectral information from conventional intensity detectors enables a superior level of material and source characterization.
在统计学中，对于一个泊松分布过程，若变量(x)以单位增量形式出现且方差为(\sigma)，则离散指数（或方差均值比）为(1)（(\sigma / \langle x\rangle = 1)）。当(x)是某种现象的度量时，该指数的值与构成该现象的量子成正比。因此，可以预期这一结果适用于极其广泛的量子现象应用测量。然而，在光子能量比例辐射探测器中，通常会观察到一组经过缩放以使量子的理想期望值为(1)的(M)个符合泊松分布的测量值({W_1, W_2, \ldots, W_M})，由于探测器损耗（如法诺在《物理评论》（1947年），第72卷，第26页中广泛指出的那样），会得出(\sigma / \langle W\rangle < 1)。在存在光谱色散的其他情况下，(\sigma / \langle W\rangle > 1)。在此，通过解析、蒙特卡罗模拟和实验研究了这些情况。这些研究揭示了一种与此类测量广泛相关的强大量子度量，其中已将其扩展到多色和有损情况。这样做表明了离散指数作为潜在量子度量的各种既定但奇怪地被忽视的作用。这项工作的X射线方面具有非常多样的用途，并已开始在射线照相和断层扫描中得到应用，从传统强度探测器中提取光谱信息的能力能够实现更高水平的材料和源表征。
