PyMieScatt > LowFrequencyMieQ() > Equation of g

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

Related (in Japanese)

• 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#L193

The implementation of g can be written as follows:

g = \frac{4}{Q_{sca}x^2}\sum(\frac{n(n+2)}{n+1}(using a_n, b_n, g1[0:4]) + \frac{2n+1}{n(n+1)}\{Re(a_n)*Re(b_n)+Im(a_n)*Im(b_n)\}

An equation similar to the above can be found:
Bohren and Huffman(1983)
p120

Q_{sca}<cos\theta> = \frac{4}{x^2}[\sum_n (\frac{n(n+2)}{n+1}Re\{a_n a_{n+1}^* + b_n b_{n+1}^* \} + \sum_n \frac{2n+1}{n(n+1)}Re\{a_n b_n^*\} ]

Since, $g = < cos\theta >$, the coefficients are same.

However, the contents of the summation seems different.

1st summation

Re(a_n a_{n+1}^* + b_n b_{n+1}^*) ... (1)

a_n a_{n+1}^*

= (Re\{a_n\} + iIm\{a_n\})(Re\{a_{n+1}\} - iIm\{ a_{n+1}\})

= Re\{a_{n}\}Re\{a_{n+1}\} - - Im\{a_{n}\}Im\{a_{n+1}\}

= Re\{a_{n}\}Re\{a_{n+1}\} + Im\{a_{n}\}Im\{a_{n+1}\} ... (2)

Eq.(1) becomes using Eq.(2)

Re(a_n a_{n+1}^* + b_n b_{n+1}^*)

= Re\{a_{n}\}Re\{a_{n+1}\} + Im\{a_{n}\}Im\{a_{n+1}\}

+ Re\{b_{n}\}Re\{b_{n+1}\} + Im\{b_{n}\}Im\{b_{n+1}\}

From above, the 1st summation is same.

2nd summation

a_n = Re\{a_n\} + i Im\{a_n\} ... (3)

b_n = Re\{a_n\} + i Im\{b_n\} ... (4)

a_n b_n^* = (Re\{a_n\} + i Im\{a_n\})(Re\{b_n\} - i Im\{b_n\})

= Re\{a_n\} Re\{b_n\} - - Im\{a_n\} Im\{b_n\}

Re\{ a_n b_n^* \} = Re\{a_n\} Re\{b_n\} + Im\{a_n\} Im\{b_n\}

From above, the second summation is same.