Delta absorbance

By definition,

${\displaystyle \Delta A=A_{\mathrm {L} }-A_{\mathrm {R} }\,}$

where ${\displaystyle \Delta A}$ (Delta Absorbance) is the difference between absorbance of left circularly polarized (LCP) and right circularly polarized (RCP) light (this is what is usually measured). ${\displaystyle \Delta A}$ is a function of wavelength, so for a measurement to be meaningful the wavelength at which it was performed must be known.

Molar circular dichroism

It can also be expressed, by applying Beer's law, as:

${\displaystyle \Delta A=(\varepsilon _{\mathrm {L} }-\varepsilon _{\mathrm {R} })Cl\,}$

where

${\displaystyle \varepsilon _{\mathrm {L} }}$ and ${\displaystyle \varepsilon _{\mathrm {R} }}$ are the molar extinction coefficients for LCP and RCP light,

${\displaystyle C}$ is the molar concentration,

${\displaystyle l}$ is the path length in centimeters (cm).

Then

${\displaystyle \Delta \varepsilon =\varepsilon _{\mathrm {L} }-\varepsilon _{\mathrm {R} }\,}$

is the molar circular dichroism. This intrinsic property is what is usually meant by the circular dichroism of the substance. Since ${\displaystyle \Delta \varepsilon }$ is a function of wavelength, a molar circular dichroism value (${\displaystyle \Delta \varepsilon }$) must specify the wavelength at which it is valid.

Extrinsic effects on circular dichroism

In many practical applications of circular dichroism (CD), as discussed below, the measured CD is not simply an intrinsic property of the molecule, but rather depends on the molecular conformation. In such a case the CD may also be a function of temperature, concentration, and the chemical environment, including solvents. In this case the reported CD value must also specify these other relevant factors in order to be meaningful.

In ordered structures lacking two-fold rotational symmetry, optical activity,^[8][9] including differential transmission^[10] (and reflection^[11]) of circularly polarized waves also depends on the propagation direction through the material. In this case, so-called extrinsic 3d chirality is associated with the mutual orientation of light beam and structure.

Molar ellipticity

Although ${\displaystyle \Delta A}$ is usually measured, for historical reasons most measurements are reported in degrees of ellipticity. Molar ellipticity is circular dichroism corrected for concentration. Molar circular dichroism and molar ellipticity, ${\displaystyle [\theta ]}$, are readily interconverted by the equation:

Elliptical polarized light (violet) is composed of unequal contributions of right (blue) and left (red) circular polarized light.

${\displaystyle [\theta ]=3298.2\,\Delta \varepsilon .\,}$

This relationship is derived by defining the ellipticity of the polarization as:

${\displaystyle \tan \theta ={\frac {E_{\mathrm {R} }-E_{\mathrm {L} }}{E_{\mathrm {R} }+E_{\mathrm {L} }}}\,}$

where

${\displaystyle E_{\mathrm {R} }}$ and ${\displaystyle E_{\mathrm {L} }}$ are the magnitudes of the electric field vectors of the right-circularly and left-circularly polarized light, respectively.

When ${\displaystyle E_{\mathrm {R} }}$ equals ${\displaystyle E_{\mathrm {L} }}$ (when there is no difference in the absorbance of right- and left-circular polarized light), ${\displaystyle \theta }$ is 0° and the light is linearly polarized. When either ${\displaystyle E_{\mathrm {R} }}$ or ${\displaystyle E_{\mathrm {L} }}$ is equal to zero (when there is complete absorbance of the circular polarized light in one direction), ${\displaystyle \theta }$ is 45° and the light is circularly polarized.

Generally, the circular dichroism effect is small, so ${\displaystyle \tan \theta }$ is small and can be approximated as ${\displaystyle \theta }$ in radians. Since the intensity or irradiance, ${\displaystyle I}$, of light is proportional to the square of the electric-field vector, the ellipticity becomes:

${\displaystyle \theta ({\text{radians}})={\frac {(I_{\mathrm {R} }^{1/2}-I_{\mathrm {L} }^{1/2})}{(I_{\mathrm {R} }^{1/2}+I_{\mathrm {L} }^{1/2})}}\,}$

Then by substituting for I using Beer's law in natural logarithm form:

${\displaystyle I=I_{0}\mathrm {e} ^{-A\ln 10}\,}$

The ellipticity can now be written as:

${\displaystyle \theta ({\text{radians}})={\frac {(\mathrm {e} ^{{\frac {-A_{\mathrm {R} }}{2}}\ln 10}-\mathrm {e} ^{{\frac {-A_{\mathrm {L} }}{2}}\ln 10})}{(\mathrm {e} ^{{\frac {-A_{\mathrm {R} }}{2}}\ln 10}+\mathrm {e} ^{{\frac {-A_{\mathrm {L} }}{2}}\ln 10})}}={\frac {\mathrm {e} ^{\Delta A{\frac {\ln 10}{2}}}-1}{\mathrm {e} ^{\Delta A{\frac {\ln 10}{2}}}+1}}\,}$

Since ${\displaystyle \Delta A\ll 1}$, this expression can be approximated by expanding the exponentials in a Taylor series to first-order and then discarding terms of ${\displaystyle \Delta A}$ in comparison with unity and converting from radians to degrees:

${\displaystyle \theta ({\text{degrees}})=\Delta A\left({\frac {\ln 10}{4}}\right)\left({\frac {180}{\pi }}\right)\,}$

The linear dependence of solute concentration and pathlength is removed by defining molar ellipticity as,

${\displaystyle [\theta ]={\frac {100\theta }{Cl}}\,}$

Then combining the last two expression with Beer's law, molar ellipticity becomes:

${\displaystyle [\theta ]=100\,\Delta \varepsilon \left({\frac {\ln 10}{4}}\right)\left({\frac {180}{\pi }}\right)=3298.2\,\Delta \varepsilon \,}$

The units of molar ellipticity are historically (deg·cm²/dmol). To calculate molar ellipticity, the sample concentration (g/L), cell pathlength (cm), and the molecular weight (g/mol) must be known.

If the sample is a protein, the mean residue weight (average molecular weight of the amino acid residues it contains) is often used in place of the molecular weight, essentially treating the protein as a solution of amino acids. Using mean residue ellipticity facilitates comparing the CD of proteins of different molecular weight; use of this normalized CD is important in studies of protein structure.

Mean residue ellipticity

Methods for estimating secondary structure in polymers, proteins and polypeptides in particular, often require that the measured molar ellipticity spectrum be converted to a normalized value, specifically a value independent of the polymer length. Mean residue ellipticity is used for this purpose; it is simply the measured molar ellipticity of the molecule divided by the number of monomer units (residues) in the molecule.