Magnetic circular dichroism (MCD) is the differential absorption of left and right circularly polarized (LCP and RCP) light, induced in a sample by a strong magnetic field oriented parallel to the direction of light propagation. MCD measurements can detect transitions which are too weak to be seen in conventional optical absorption spectra, and it can be used to distinguish between overlapping transitions. Paramagnetic systems are common analytes, as their near-degenerate magnetic sublevels provide strong MCD intensity that varies with both field strength and sample temperature. The MCD signal also provides insight into the symmetry of the electronic levels of the studied systems, such as metal ion sites.
History.
It was first shown by Faraday that optical activity (the Faraday effect) could be induced in matter by a longitudinal magnetic field (a field in the direction of light propagation). The development of MCD really began in the 1930s when a quantum mechanical theory of MOR (magnetic optical rotatory dispersion) in regions outside absorption bands was formulated. The expansion of the theory to include MCD and MOR effects in the region of absorptions, which were referred to as "anomalous dispersions" was developed soon thereafter. There was, however, little effort made to refine MCD as a modern spectroscopic technique until the early 1960s. Since that time there have been numerous studies of MCD spectra for a very large variety of samples, including stable molecules in solutions, in isotropic solids, and in the gas phase, as well as unstable molecules entrapped in noble gas matrices. More recently, MCD has found useful application in the study of biologically important systems including metalloenzymes and proteins containing metal centers.
Differences between CD and MCD.
In natural optical activity, the difference between the LCP light and the RCP light is caused by the asymmetry of the molecules (i.e. chiral molecules). Because of the handedness of the molecule, the absorption of the LCP light would be different from the RCP light. However, in MCD in the presence of a magnetic field, LCP and RCP no longer interact equivalently with the absorbing medium. Thus, there is not the same direct relation between magnetic optical activity and molecular stereochemistry which would be expected, because it is found in natural optical activity. So, natural CD is much more rare than MCD which does not strictly require the target molecule to be chiral.
Although there is much overlap in the requirements and use of instruments, ordinary CD instruments are usually optimized for operation in the ultraviolet, approximately 170–300 nm, while MCD instruments are typically required to operate in the visible to near infrared, approximately 300–2000 nm. The physical processes that lead to MCD are substantively different from those of CD. However, like CD, it is dependent on the differential absorption of left and right hand circularly polarized light. MCD will only exist at a given wavelength if the studied sample has an optical absorption at that wavelength. This is distinctly different from the related phenomenon of optical rotatory dispersion (ORD), which can be observed at wavelengths far from any absorption band.
Measurement.
The MCD signal ΔA is derived via the absorption of the LCP and RCP light as
formula_1
This signal is often presented as a function of wavelength λ, temperature T or magnetic field H. MCD spectrometers can simultaneously measure absorbance and ΔA along the same light path. This eliminates error introduced through multiple measurements or different instruments that previously occurred before this advent.
The MCD spectrometer example shown below begins with a light source that emits a monochromatic wave of light. This wave is passed through a Rochon prism linear polarizer, which separates the incident wave into two beams that are linearly polarized by 90 degrees. The two beams follow different paths- one beam (the extraordinary beam) traveling directly to a photomultiplier (PMT), and the other beam (the ordinary beam) passing through a photoelastic modulator (PEM) oriented at 45 degrees to the direction of the ordinary ray polarization. The PMT for the extraordinary beam detects the light intensity of the input beam. The PEM is adjusted to cause an alternating plus and minus 1/4 wavelength shift of one of the two orthogonal components of the ordinary beam. This modulation converts the linearly polarized light into circularly polarized light at the peaks of the modulation cycle. Linearly polarized light can be decomposed into two circular components with intensity represented as formula_2
The PEM will delay one component of linearly polarized light with a time dependence that advances the other component by 1/4 λ (hence, quarter-wave shift). The departing circularly polarized light oscillates between RCP and LCP in a sinusoidal time-dependence as depicted below:
The light finally travels through a magnet containing the sample, and the transmittance is recorded by another PMT. The schematic is given below:
The intensity of light from the ordinary wave that reaches the PMT is governed by the equation:
formula_3
Here A– and A+ are the absorbances of LCP or RCP, respectively; ω is the modulator frequency – usually a high acoustic frequency such as 50 kHz; "t" is time; and δ0 is the time-dependent wavelength shift.
This intensity of light passing through the sample is converted into a two-component voltage via a current/voltage amplifier. A DC voltage will emerge corresponding to the intensity of light passed through the sample. If there is a ΔA, then a small AC voltage will be present that corresponds to the modulation frequency, ω. This voltage is detected by the lock in amplifier, which receives its reference frequency, ω, directly from the PEM. From such voltage, ΔA and A can be derived using the following relations:
formula_4
formula_5
where Vex is the (DC) voltage measured by the PMT from the extraordinary wave, and Vdc is the DC component of the voltage measured by the PMT for the ordinary wave (measurement path not shown in the diagram).
Some superconducting magnets have a small sample chamber, far too small to contain the entire optical system. Instead, the magnet sample chamber has windows on two opposite sides. Light from the source enters one side, interacts with the sample (usually also temperature controlled) in the magnetic field, and exits through the opposite window to the detector. Optical relay systems that allow the source and detector each to be about a meter from the sample are typically employed. This arrangement avoids many of the difficulties that would be encountered if the optical apparatus had to operate in the high magnetic field, and also allows for a much less expensive magnet.
Applications.
MCD can be used as an optical technique for the detection of electronic structure of both the ground states and excited states. It is also a strong addition to the more commonly used absorption spectroscopy, and there are two reasons that explain this. First, a transition buried under a stronger transition can appear in MCD if the first derivative of the absorption is much larger for the weaker transition or it is of the opposite sign. Second, MCD will be found where no absorption is detected at all if ΔA ; (ΔAmin) but A ; Amin, where (ΔA)min and Amin are the minimum of ΔA and A that are detectable. Typically, (ΔAmin) and Amin are of the magnitudes around 10−5 and 10−3 respectively. So, a transition can only be detected in MCD, not in the absorption spectroscopy, if ΔA/A ; 10−2. This happens in paramagnetic systems that are at lower temperature or that have sharp lines in the spectroscopy.
In biology, metalloproteins are the most likely candidates for MCD measurements, as the presence of metals with degenerate energy levels leads to strong MCD signals. In the case of ferric heme proteins, MCD is capable of determining both oxidation and spin state to a remarkably exquisite degree. In regular proteins, MCD is capable of stoichiometrically measuring the tryptophan content of proteins, assuming there are no other competing absorbers in the spectroscopic system.
In addition, the application of MCD spectroscopy greatly improved the level of understanding in the ferrous non-heme systems because of the direct observation of the d–d transitions, which generally can not be obtained in optical absorption spectroscopy owing to the weak extinction coefficients and are often electron paramagnetic resonance silent due to relatively large ground-state sublevel splittings and fast relaxation times.
Theory.
Consider a system of localized, non-interacting absorbing centers. Based on the semi-classical radiation absorption theory within the electric dipole approximation, the electric vector of the circularly polarized waves propagates along the +z direction. In this system, formula_6 is the angular frequency, and formula_7 = n – ik is the complex refractive index. As the light travels, the attenuation of the beam is expressed as
formula_8 
where formula_9 is the intensity of light at position formula_10, formula_11 is the absorption coefficient of the medium in the formula_10 direction, and formula_13 is the speed of light. Circular dichroism (CD) is then defined by the difference between left (formula_14) and right (formula_15) circularly polarized light, formula_16, following the sign convention of natural optical activity. In the presence of a static, uniform external magnetic field applied parallel to the direction of propagation of light, the Hamiltonian for the absorbing center takes the form formula_17 for formula_18 describing the system in the external magnetic field and formula_19 describing the applied electromagnetic radiation. The absorption coefficient for a transition between two eigenstates of formula_18, formula_21 and formula_22, can be described using the electric dipole transition operator formula_23 as 
formula_24
formula_25
The formula_26 term is a frequency-independent correction factor allowing for the effect of the medium on the light wave electric field, composed of the permittivity formula_27 and the real refractive index formula_28.
Discrete line spectrum.
In cases of a discrete spectrum, the observed formula_29 at a particular frequency formula_30 can be treated as a sum of contributions from each transition, 
formula_31 
where formula_32 is the contribution at formula_30 from the formula_34 transition, formula_35 is the absorption coefficient for the formula_34 transition, and formula_37 is a bandshape function (formula_38). Because eigenstates formula_21 and formula_22 depend on the applied external field, the value of formula_41 varies with field. It is frequently useful to compare this value to the absorption coefficient in the absence of an applied field, often denoted 
formula_42 
When the Zeeman effect is small compared to zero-field state separations, line width, and formula_43 and when the line shape is independent of the applied external field formula_44, first-order perturbation theory can be applied to separate formula_29 into three contributing Faraday terms, called formula_46, formula_47, and formula_48. The subscript indicates the moment such that formula_46 contributes a derivative-shaped signal and formula_47 and formula_48 contribute regular absorptions. Additionally, a zero-field absorption term formula_52 is defined. The relationships between formula_29, formula_54, and these Faraday terms are 
formula_55
formula_56 
for external field strength formula_44, Boltzmann constant formula_58, temperature formula_59, and a proportionality constant formula_60. This expression requires assumptions that formula_22 is sufficiently high in energy that formula_62, and that the temperature of the sample is high enough that magnetic saturation does not produce nonlinear formula_63 term behavior. 
Though one must pay attention to proportionality constants, there is a proportionality between formula_29 and molar extinction coefficient formula_65 and absorbance formula_66 for concentration formula_67 and path length formula_68. 
These Faraday terms are the usual language in which MCD spectra are discussed. Their definitions from perturbation theory are 
formula_69 
where formula_70 is the degeneracy of ground state formula_71, formula_72 labels states other than formula_71 or formula_74, formula_27 and formula_76 and formula_77 label the levels within states formula_71 and formula_74 and formula_72 (respectively), formula_81 is the energy of unperturbed state formula_82, formula_83 is the formula_10 angular momentum operator, formula_85 is the formula_10 spin operator, and formula_87 indicates the real part of the expression. 
Origins of A, B, and C Faraday Terms.
The equations in the previous subsection reveal that the formula_46, formula_47, and formula_48 terms originate through three distinct mechanisms. 
The formula_46 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 formula_47 term is due to the field-induced mixing of states. Energetic proximity of a third state formula_93 to either the ground state formula_94 or excited state formula_95 gives appreciable Zeeman coupling in the presence of an applied external field. As the strength of the magnetic field increases,