### Abstract

A new approach to the use of spectroscopic absorbance measurements for anisotropic crystals allows results to be extracted using unpolarized light incident on random crystal orientations. The theory of light propagation in anisotropic absorbing crystals is developed from Maxwell's equations to devise an expression for the transmittance of linearly polarized light traveling in an arbitrary direction in weakly absorbing media. This theory predicts the distribution of transmittance and absorbance as a function of direction and polarization angle of incident light. It is shown how a previously deduced empirical expression, commonly used in infrared spectroscopy, is a good approximation to the full theory under a wide range of conditions. The new theory shows that principal polarized absorbances correspond to the eigenvalues of an absorbance ellipsoid. An expression is derived for the unpolarized absorbance as a function of the angles describing incident light direction, A_{unpol}(φ, ψ), and the principal polarized absorbances, A_{a}, A_{b}, A_{c} in an anisotropic crystal. A_{unpol}(φ,ψ) = 1/2[A_{a}(cos^{2}φcos^{2}ψ + sin^{2}ψ) + A_{b}(cos^{2}φsin^{2}ψ + cos^{2}ψ) + A_{c}sin^{2}φ]. Integration of this expression over all incident angles leads to a simple relationship between total measured unpolarized absorbance and the three principal polarized absorbances. Using this theory, a procedure is proposed for estimating both total (A_{a} + A_{b} + A_{c}) and principal absorbances from spectroscopic measurements of absorbance using unpolarized light on a set of randomly oriented crystals.

Original language | English |
---|---|

Pages (from-to) | 751-764 |

Number of pages | 14 |

Journal | American Mineralogist |

Volume | 93 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - May 2008 |

### Fingerprint

### Keywords

- Absorption index theory
- Anisotropic media
- Spectroscopy
- Unpolarized light

### ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics

### Cite this

*American Mineralogist*,

*93*(5-6), 751-764. https://doi.org/10.2138/am.2008.2657

**Quantitative absorbance spectroscopy with unpolarized light : Part I. Physical and mathematical development.** / Sambridge, Malcolm; Gerald, John Fitz; Kovács, I.; O'Neill, Hugh St C; Hermann, Jörg.

Research output: Contribution to journal › Article

*American Mineralogist*, vol. 93, no. 5-6, pp. 751-764. https://doi.org/10.2138/am.2008.2657

}

TY - JOUR

T1 - Quantitative absorbance spectroscopy with unpolarized light

T2 - Part I. Physical and mathematical development

AU - Sambridge, Malcolm

AU - Gerald, John Fitz

AU - Kovács, I.

AU - O'Neill, Hugh St C

AU - Hermann, Jörg

PY - 2008/5

Y1 - 2008/5

N2 - A new approach to the use of spectroscopic absorbance measurements for anisotropic crystals allows results to be extracted using unpolarized light incident on random crystal orientations. The theory of light propagation in anisotropic absorbing crystals is developed from Maxwell's equations to devise an expression for the transmittance of linearly polarized light traveling in an arbitrary direction in weakly absorbing media. This theory predicts the distribution of transmittance and absorbance as a function of direction and polarization angle of incident light. It is shown how a previously deduced empirical expression, commonly used in infrared spectroscopy, is a good approximation to the full theory under a wide range of conditions. The new theory shows that principal polarized absorbances correspond to the eigenvalues of an absorbance ellipsoid. An expression is derived for the unpolarized absorbance as a function of the angles describing incident light direction, Aunpol(φ, ψ), and the principal polarized absorbances, Aa, Ab, Ac in an anisotropic crystal. Aunpol(φ,ψ) = 1/2[Aa(cos2φcos2ψ + sin2ψ) + Ab(cos2φsin2ψ + cos2ψ) + Acsin2φ]. Integration of this expression over all incident angles leads to a simple relationship between total measured unpolarized absorbance and the three principal polarized absorbances. Using this theory, a procedure is proposed for estimating both total (Aa + Ab + Ac) and principal absorbances from spectroscopic measurements of absorbance using unpolarized light on a set of randomly oriented crystals.

AB - A new approach to the use of spectroscopic absorbance measurements for anisotropic crystals allows results to be extracted using unpolarized light incident on random crystal orientations. The theory of light propagation in anisotropic absorbing crystals is developed from Maxwell's equations to devise an expression for the transmittance of linearly polarized light traveling in an arbitrary direction in weakly absorbing media. This theory predicts the distribution of transmittance and absorbance as a function of direction and polarization angle of incident light. It is shown how a previously deduced empirical expression, commonly used in infrared spectroscopy, is a good approximation to the full theory under a wide range of conditions. The new theory shows that principal polarized absorbances correspond to the eigenvalues of an absorbance ellipsoid. An expression is derived for the unpolarized absorbance as a function of the angles describing incident light direction, Aunpol(φ, ψ), and the principal polarized absorbances, Aa, Ab, Ac in an anisotropic crystal. Aunpol(φ,ψ) = 1/2[Aa(cos2φcos2ψ + sin2ψ) + Ab(cos2φsin2ψ + cos2ψ) + Acsin2φ]. Integration of this expression over all incident angles leads to a simple relationship between total measured unpolarized absorbance and the three principal polarized absorbances. Using this theory, a procedure is proposed for estimating both total (Aa + Ab + Ac) and principal absorbances from spectroscopic measurements of absorbance using unpolarized light on a set of randomly oriented crystals.

KW - Absorption index theory

KW - Anisotropic media

KW - Spectroscopy

KW - Unpolarized light

UR - http://www.scopus.com/inward/record.url?scp=41749113486&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41749113486&partnerID=8YFLogxK

U2 - 10.2138/am.2008.2657

DO - 10.2138/am.2008.2657

M3 - Article

AN - SCOPUS:41749113486

VL - 93

SP - 751

EP - 764

JO - American Mineralogist

JF - American Mineralogist

SN - 0003-004X

IS - 5-6

ER -