We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain ${\mathbb{R}}^d$ for arbitrary $d\ge 1.$ A continuous family of pseudo-problems and related H functions arises and includes the classical 3D solutions, as well as 2D “Flatland” and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension d are highlighted and we find that a discrete diffusion mode is not universal for d > 3 in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.

我们解决了各向同性散射半空间的平面平行照明在推广到欧几里得域时的经典反照率和Milne问题 ${\mathbb{[R}}^d$ 对于任意 $d\ge \mathrm{1个}。$出现了一系列连续的伪问题和相关的H函数，其中包括经典3D解决方案以及2D“平地”和拉杆模型解决方案（作为特殊情况）。案例式特征模式方法应用于一般问题，内部标量密度，新兴分布及其各自的矩均以封闭形式表示。突出了尺寸d不变的通用属性，我们发现对于d  > 3的吸收介质，离散扩散模式不是通用的。我们还发现不同维度之间以及高维度各向异性3D散射和各向同性散射之间的意外对应关系。