PyMieScatt > LowFrequencyMieQ() > Equation of Qsca

動作環境

GeForce GTX 1070 (8GB)
ASRock Z170M Pro4S [Intel Z170chipset]
Ubuntu 16.04 LTS desktop amd64
TensorFlow v1.2.1
cuDNN v5.1 for Linux
CUDA v8.0
Python 3.5.2
IPython 6.0.0 -- An enhanced Interactive Python.
gcc (Ubuntu 5.4.0-6ubuntu1~16.04.4) 5.4.0 20160609
GNU bash, version 4.3.48(1)-release (x86_64-pc-linux-gnu)
scipy v0.19.1
geopandas v0.3.0
MATLAB R2017b (Home Edition)
ADDA v.1.3b6
gnustep-gui-runtime v0.24.0-3.1

関連

• PyMieScatt > LowFrequencyMieQ() > for small size parameter
• PyMieScatt > LowFrequencyMieQ() > Equation of Qext | 新人応援 > １次情報と実装の乖離

Reference

• Bohren C.F. and D. R. Huffman, "Absorption and Scattering of Light by Small Particles", John Wiley, New York, NY (1983)
• PDF: MATLAB Functions for Mie Scattering and Absorption, Research Report No. 2002-08, June 2002 by Christian Mätzler

Equation of Qsca

http://pymiescatt.readthedocs.io/en/latest/forward.html#LowFrequencyMieQ
https://github.com/bsumlin/PyMieScatt/blob/master/PyMieScatt/Mie.py#L175

LowFrequencyMieQ()のQscaの実装を式に起こすと以下となる。

Q_{sca} = \frac{2}{x^2}\sum(2n+1)\{|a_n|^2 + |b_n|^2\}

この式はどこから来たのか?

Bohren and Huffman (1983)
p103

C_{sca} = \frac{W_s}{I_i} = \frac{2\pi}{k^2}\sum_{n=1}^{\infty}(2n+1)(|a_n|^2 + |b_n|^2) ... (4.61)

Efficiency QとCross section Cには以下の関係式がある。

Q_{sca} = \frac{C_{sca}}{\pi a^2} ... (1)

ここで、aは粒子の半径。

Eq.(4.61)とEq.(1)より

Q_{sca} = \frac{2}{k^2 a^2}\sum_{n=1}^{\infty}(2n+1)(|a_n|^2 + |b_n|^2) ... (2)

size parameter Xは以下として定義される (Bohren and Huffman(1983) p100 (Eq. 4.51の下の式)。

x = ka = \frac{2\pi Na}{\lambda} ... (3)

Eq. (2)と Eq.(3)より

Q_{sca} = \frac{2}{x^2}\sum(2n+1)\{|a_n|^2 + |b_n|^2\}

導出完了。