Given a value of the Fermi-Dirac integral $\underset{0}{\overset{\infty }{\int }}\frac{t^{j}dt}{e^{t-\eta }+1}$, this routine returns the value of the parameter η for a given j with a relative error less than ${10}^{-13}$. The inversion is provided for $\eta \in [-30,100]$ and for the following set of j values, $\{-1/2,1/2,3/2,5/2,1,2\}$. For $\eta \in [-30,-2]$, an iterative method involving the McDougall and Stoner series is used. For the rest of the region of interest, a rational fit is employed.

Program summary

Program title: InvFD

CPC Library link to program files: https://doi.org/10.17632/rc8x92mx8n.1

Licensing provisions: GPLv3

Programming language: Octave-4.2.2

Nature of problem: Quite often in solid-state physics applications, given a value of the Fermi-Dirac integral defined by $\underset{0}{\overset{\infty }{\int }}\frac{t^{j}dt}{e^{t-\eta }+1}$, one needs the value of the parameter η, for a given j where $j\in \{-1/2,1/2,3/2,5/2,1,2\}$. The η values considered lie in the range $[-30,100]$, a typical region of interest for solidstate physics applications.

Solution method: For $\eta \in [-30,-2]$, an iterative scheme based on the McDougall-Stoner series is utilized. For the remaining interval $[-2,100]$, a rational fitting is provided. The relative error of the η evaluation is less than ${10}^{-13}$.

Additional comments including restrictions and unusual features: Provides inversion for both integer and half-integer Fermi-Dirac integrals. The relative accuracy is close to double precision.

中文翻译：

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InvFD，用于费米-狄拉克积分数值反演的 OCTAVE 例程

给定 Fermi-Dirac 积分的值 $\underset{0}{\overset{\infty }{\int }}\frac{{吨}^{j}d吨}{{\mathrm{电子}}^{吨-\eta }+1}$，此例程返回给定j的参数η的值，其相对误差小于${10}^{-13}$. 反演是为$\eta \in [-30,100]$ 对于以下一组 j 值， $\{-1/2,1/2,3/2,5/2,1,2\}$. 为了$\eta \in [-30,-2]$，使用涉及 McDougall 和 Stoner 系列的迭代方法。对于感兴趣区域的其余部分，采用合理拟合。

程序概要

程序名称： InvFD

CPC 库程序文件链接： https : //doi.org/10.17632/rc8x92mx8n.1

许可条款： GPLv3

编程语言： Octave-4.2.2

问题的性质：在固态物理应用中很常见，给定一个由下式定义的费米-狄拉克积分值$\underset{0}{\overset{\infty }{\int }}\frac{{吨}^{j}d吨}{{\mathrm{电子}}^{吨-\eta }+1}$，对于给定的j，需要参数η的值，其中$j\in \{-1/2,1/2,3/2,5/2,1,2\}$. 考虑的η值在范围内$[-30,100]$，固态物理应用的典型感兴趣区域。

解决方法：对于$\eta \in [-30,-2]$，使用基于 McDougall-Stoner 系列的迭代方案。对于剩余间隔$[-2,100]$，提供了合理的拟合。η评估的相对误差小于${10}^{-13}$.

包括限制和不寻常功能在内的其他评论：提供整数和半整数费米-狄拉克积分的反演。相对精度接近双精度。