论文标题
$ n $尺寸的Madelung常数
The Madelung Constant in $N$ Dimensions
论文作者
论文摘要
我们介绍了两个收敛的系列扩展（直接和递归），以$ n $ dipersional madelung常数为$ r_n（m）$的$ r_n（m）$的表示，$ m_n（s）$，其中$ s $是Madelung系列的指数（通常选择为$ s = 1/2 $）。详细讨论了包括分析延续的功能行为以及贝塞尔功能扩展的收敛性。递归定义用于评估$ r_n（m）$。 $ m_n（s）$的$ s = \ tfrac {1} {2}，\ tfrac {3} {2} {2}，3 $和3 $和6的尺寸，最高$ n = 20 $，以及$ m_n（1/2）$ to $ n = $ n = 100 $。扎克（Zucker）对$ n $ dimensional Madelung常数的原始分析，即使尺寸最高为$ n = 8 $，并且可能会简要分析其可能的延续到更高维度。
We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung series (usually chosen as $s=1/2$). The functional behavior including analytical continuation, and the convergence of the Bessel function expansion is discussed in detail. Recursive definitions are used to evaluate $r_N(m)$. Values for $M_N(s)$ for $s=\tfrac{1}{2}, \tfrac{3}{2}, 3$ and 6 for dimension up to $N=20$ and for $M_N(1/2)$ up to $N=100$ are presented. Zucker's original analysis on $N$-dimensional Madelung constants for even dimensions up to $N=8$ and their possible continuation into higher dimensions is briefly analyzed.
