Bloch space

LBOZ is a coefficient used in spectrophotometry to estimate selectivityTemplate:Dn (amount of overlappingTemplate:Dn of spectra) in quantitative manner. It is named after its creators: Lorber, Bergmann, von Oepen, and Zinn.

Definition

Sample synthetic spectra of four components

Let $\mathbf {X}$ be a matrix of the spectra (absorbances), where the k rows correspond to the components in mixture and n columns correspond to the sequence of wavelengths. The LBOZ criterion for kth component is calculated from the following formula:

$\xi _{k}={\frac {1}{\|\mathbf {X} _{k-row}\|\|\mathbf {X} _{k-col}^{+}\|}}$

where $\mathbf {X} ^{+}$ means a pseudoinverse of the matrix and $\|\cdots \|$ means an euclidean length of a vector.

Properties

The image above show synthetic gaussian spectra. The LBOZ criteria are: 0.561 for black compound, 0.402 for red compound, 0.899 for green and 0.549 for blue. LBOZ always lie in range <0,1> and has strong mathematical sense - it presents the amount of spectral signal which is not overlapped by the others. Hence, the uncertainty of a compound quantity increases by $1/\xi$ in presence of the other compounds. In this case, the highest uncertainty is expected during determination of red compound - theoretically 2.38 times greater than during determination of its compound alone.

References

• A. Lorber, Anal. Chem. 58 (1986) 1167.
• A. Lorber, A. Harel, Z. Goldbart, I.B. Brenner, Anal. Chem. 59 (1987) 1260.
• G. Bergmann, B. von Oepen, P. Zinn, Anal. Chem. 59 (1987) 2522.