Discrete line spectrum

In cases of a discrete spectrum, the observed ${\displaystyle \Delta k}$ at a particular frequency ${\displaystyle \omega }$ can be treated as a sum of contributions from each transition,

${\displaystyle \Delta k_{\mathrm {obs} }(\omega )=\sum _{a,j}\Delta k_{a\to j}(\omega )=\sum _{a,j}[\Delta k_{a\to j}]f_{ja}(\omega )}$

where ${\displaystyle \Delta k_{a\to j}(\omega )}$ is the contribution at ${\displaystyle \omega }$ from the ${\displaystyle a\to j}$ transition, ${\displaystyle [\Delta k_{a\to j}]}$ is the absorption coefficient for the ${\displaystyle a\to j}$ transition, and ${\displaystyle f_{ja}(\omega )}$ is a bandshape function (${\displaystyle \textstyle {\int _{0}^{\infty }f_{ja}(\omega )d\omega =1}}$). Because eigenstates ${\displaystyle a}$ and ${\displaystyle j}$ depend on the applied external field, the value of ${\displaystyle \Delta k_{\mathrm {obs} }(\omega )}$ varies with field. It is frequently useful to compare this value to the absorption coefficient in the absence of an applied field, often denoted

${\displaystyle k^{0}(\omega )=\sum _{a,j}k_{a\to j}^{0}(\omega )=\sum _{a,j}[k_{a\to j}^{0}]f_{ja}^{0}(\omega )}$

When the Zeeman effect is small compared to zero-field state separations, line width, and ${\displaystyle kT}$ and when the line shape is independent of the applied external field ${\displaystyle H}$, first-order perturbation theory can be applied to separate ${\displaystyle \Delta k}$ into three contributing Faraday terms, called ${\displaystyle {\mathcal {A}}_{1}}$, ${\displaystyle {\mathcal {B}}_{0}}$, and ${\displaystyle {\mathcal {C}}_{0}}$. The subscript indicates the moment such that ${\displaystyle {\mathcal {A}}_{1}}$ contributes a derivative-shaped signal and ${\displaystyle {\mathcal {B}}_{0}}$ and ${\displaystyle {\mathcal {C}}_{0}}$ contribute regular absorptions. Additionally, a zero-field absorption term ${\displaystyle {\mathcal {D}}_{0}}$ is defined. The relationships between ${\displaystyle \Delta k}$, ${\displaystyle k^{0}}$, and these Faraday terms are

${\displaystyle \Delta k_{A\to J}(\omega )=-{\frac {4}{3}}\gamma N_{A}^{0}\left\{{\frac {{\mathcal {A}}_{1}(A\to J)}{\hbar }}{\frac {\partial f_{ja}^{0}(\omega )}{\partial \omega }}+\left[{\mathcal {B}}_{0}(A\to J)+{\frac {{\mathcal {C}}_{0}(A\to J)}{k_{B}T}}\right]f_{ja}^{0}(\omega )\right\}H}$

${\displaystyle k_{A\to J}^{0}(\omega )={\frac {2}{3}}\gamma N_{A}^{0}{\mathcal {D}}_{0}(A\to J)f_{ja}^{0}(\omega )}$

for external field strength ${\displaystyle H}$, Boltzmann constant ${\displaystyle k_{B}}$, temperature ${\displaystyle T}$, and a proportionality constant ${\displaystyle \gamma }$. This expression requires assumptions that ${\displaystyle j}$ is sufficiently high in energy that ${\displaystyle N_{j}\approx 0}$, and that the temperature of the sample is high enough that magnetic saturation does not produce nonlinear ${\displaystyle {\mathcal {C}}}$ term behavior. Though one must pay attention to proportionality constants, there is a proportionality between ${\displaystyle \Delta k}$ and molar extinction coefficient ${\displaystyle \epsilon }$ and absorbance ${\displaystyle A/Cl}$ for concentration ${\displaystyle C}$ and path length ${\displaystyle l}$.

These Faraday terms are the usual language in which MCD spectra are discussed. Their definitions from perturbation theory are^[10]

${\displaystyle {\begin{aligned}{\mathcal {A}}_{1}&=-{\frac {1}{d_{A}}}\sum _{\alpha ,\lambda }\left(\langle J_{\lambda }|L_{z}+2S_{z}|J_{\lambda }\rangle -\langle A_{\alpha }|L_{z}+2S_{z}|A_{\alpha }\rangle \right)\times \left(|\langle A_{\alpha }|m_{-}|J_{\lambda }\rangle |^{2}-\langle A_{\alpha }|m_{+}|J_{\lambda }\rangle |^{2}\right)\\{\mathcal {B}}_{0}&={\frac {2}{d_{A}}}\Re \sum _{\alpha ,\lambda }\left[\sum _{K\neq J,\kappa }{\frac {1}{E_{K}-E_{J}}}\langle J_{\lambda }|L_{z}+2S_{z}|K_{\kappa }\rangle \times \left(\langle A_{\alpha }|m_{-}|J_{\lambda }\rangle \langle K_{\kappa }|m_{+}|A_{\alpha }\rangle -\langle A_{\alpha }|m_{+}|J_{\lambda }\rangle \langle K_{\kappa }|m_{-}|A_{\alpha }\rangle \right)\right.\\&\qquad \left.+\sum _{K\neq A,\kappa }{\frac {1}{E_{K}-E_{A}}}\langle K_{\kappa }|L_{z}+2S_{z}|A_{\alpha }\rangle \times \left(\langle A_{\alpha }|m_{-}|J_{\lambda }\rangle \langle J_{\lambda }|m_{+}|K_{\kappa }\rangle -\langle A_{\alpha }|m_{+}|J_{\lambda }\rangle \langle J_{\lambda }|m_{-}|K_{\kappa }\rangle \right)\right]\\{\mathcal {C}}_{0}&={\frac {1}{d_{A}}}\sum _{\alpha ,\lambda }\langle A_{\alpha }|L_{z}+2S_{z}|A_{\alpha }\rangle \times \left(|\langle A_{\alpha }|m_{-}|J_{\lambda }\rangle |^{2}-\langle A_{\alpha }|m_{+}|J_{\lambda }\rangle |^{2}\right)\\{\mathcal {D}}_{0}&={\frac {1}{2d_{A}}}\sum _{\alpha ,\lambda }\left(|\langle A_{\alpha }|m_{-}|J_{\lambda }\rangle |^{2}+\langle A_{\alpha }|m_{+}|J_{\lambda }\rangle |^{2}\right)\end{aligned}}}$

where ${\displaystyle d_{A}}$ is the degeneracy of ground state ${\displaystyle A}$, ${\displaystyle K}$ labels states other than ${\displaystyle A}$ or ${\displaystyle J}$, ${\displaystyle \alpha }$ and ${\displaystyle \lambda }$ and ${\displaystyle \kappa }$ label the levels within states ${\displaystyle A}$ and ${\displaystyle J}$ and ${\displaystyle K}$ (respectively), ${\displaystyle E_{X}}$ is the energy of unperturbed state ${\displaystyle X}$, ${\displaystyle L_{z}}$ is the ${\displaystyle z}$ angular momentum operator, ${\displaystyle S_{z}}$ is the ${\displaystyle z}$ spin operator, and ${\displaystyle \Re }$ indicates the real part of the expression.

Origins of A, B, and C Faraday Terms

${\displaystyle {\mathcal {A}}_{1}}$, ${\displaystyle {\mathcal {B}}_{0}}$, and ${\displaystyle {\mathcal {C}}_{0}}$ term intensity mechanisms for magnetic circular dichroism (MCD) signal

The equations in the previous subsection reveal that the ${\displaystyle {\mathcal {A}}_{1}}$, ${\displaystyle {\mathcal {B}}_{0}}$, and ${\displaystyle {\mathcal {C}}_{0}}$ terms originate through three distinct mechanisms.

The ${\displaystyle {\mathcal {A}}_{1}}$ term arises from Zeeman splitting of the ground or excited degenerate states. These field-dependent changes in energies of the magnetic sublevels causes small shifts in the bands to higher/lower energy. The slight offsets result in incomplete cancellation of the positive and negative features, giving a net derivative shape in the spectrum. This intensity mechanism is generally independent of sample temperature.

The ${\displaystyle {\mathcal {B}}_{0}}$ term is due to the field-induced mixing of states. Energetic proximity of a third state ${\displaystyle |K\rangle }$ to either the ground state ${\displaystyle |A\rangle }$ or excited state ${\displaystyle |J\rangle }$ gives appreciable Zeeman coupling in the presence of an applied external field. As the strength of the magnetic field increases, the amount of mixing increases to give growth of an absorption band shape. Like the ${\displaystyle {\mathcal {A}}_{1}}$ term, the ${\displaystyle {\mathcal {B}}_{0}}$ term is generally temperature independent. Temperature dependence of ${\displaystyle {\mathcal {B}}_{0}}$ term intensity can sometimes be observed when ${\displaystyle |K\rangle }$ is particularly low-lying in energy.

The ${\displaystyle {\mathcal {C}}_{0}}$ term requires the degeneracy of the ground state, often encountered for paramagnetic samples. This happens due to a change in the Boltzmann population of the magnetic sublevels, which is dependent on the degree of field-induced splitting of the sublevel energies and on the sample temperature.^[11] Decrease of the temperature and increase of the magnetic field increases the ${\displaystyle {\mathcal {C}}_{0}}$ term intensity until it reaches the maximum (saturation limit). Experimentally, the ${\displaystyle {\mathcal {C}}_{0}}$ term spectrum can be obtained from MCD raw data by subtraction of MCD spectra measured in the same applied magnetic field at different temperatures, while ${\displaystyle {\mathcal {A}}_{1}}$ and ${\displaystyle {\mathcal {B}}_{0}}$ terms can be distinguished via their different band shapes.^[9]

The relative contributions of A, B and C terms to the MCD spectrum are proportional to the inverse line width, energy splitting, and temperature:

${\displaystyle A:B:C={\frac {1}{\Delta \Gamma }}:{\frac {1}{\Delta E}}:{\frac {1}{kT}}}$

where ${\displaystyle \Delta \Gamma }$ is line width and ${\displaystyle \Delta E}$ is the zero-field state separation. For typical values of ${\displaystyle \Delta \Gamma }$ = 1000 cm⁻¹, ${\displaystyle \Delta E}$ = 10,000 cm⁻¹ and ${\displaystyle kT}$ = 6 cm⁻¹ (at 10 K), the three terms make relative contributions 1:0.1:150. So, at low temperature the ${\displaystyle {\mathcal {C}}_{0}}$ term dominates over ${\displaystyle {\mathcal {A}}_{1}}$ and ${\displaystyle {\mathcal {B}}_{0}}$ for paramagnetic samples.^[12]