The Hamiltonian for the system

The Hamiltonian for the two-spin system mentioned above can be described as

${\displaystyle {\mathcal {H}}_{\mathrm {0} }={\mathcal {H}}_{\mathrm {EZ} }+{\mathcal {H}}_{\mathrm {NZ} }+{\mathcal {H}}_{\mathrm {HFS} }+{\mathcal {H}}_{\mathrm {Q} }}$

The four terms in this equation describe the electron Zeeman interaction (EZ), the nuclear Zeeman interaction (NZ), the hyperfine interaction (HFS), and the nuclear quadrupole interaction (Q), respectively.^[4]

The electron Zeeman interaction describes the interaction between an electron spin and the applied magnetic field. The nuclear Zeeman interaction is the interaction of the magnetic moment of the proton with an applied magnetic field. The hyperfine interaction is the coupling between the electron spin and the proton's nuclear spin. The nuclear quadrupole interaction is present only in nuclei with I>1/2.

ENDOR spectra contain information on the type of nuclei in the vicinity of the unpaired electron (NZ and EZ), on the distances between nuclei and on the spin density distribution (HFS) and on the electric field gradient at the nuclei (Q).

Principle of the ENDOR method

Energy level diagram for the ENDOR method

The right figure illustrates the energy diagram of the simplest spin system where a is the isotropic hyperfine coupling constant in hertz (Hz). This diagram indicates the electron Zeeman, nuclear Zeeman and hyperfine splittings. In a steady state ENDOR experiment, an EPR transition (A, D), called the observer, is partly saturated by microwave radiation of amplitude ${\displaystyle \mathrm {B} _{\mathrm {1} }}$ while a driving radio frequency (rf) field of amplitude ${\displaystyle \mathrm {B} _{\mathrm {2} }}$, called the pump, induces nuclear transitions.^[5] Transitions happen at frequencies ${\displaystyle \nu _{\mathrm {1} }}$ and ${\displaystyle \nu _{\mathrm {2} }}$ and obey the NMR selection rules ${\displaystyle \Delta M_{I}=\pm 1}$ and ${\displaystyle \Delta M_{S}=0}$. It is these NMR transitions that are detected by ENDOR via the intensity changes to the simultaneously irradiated EPR transition. It is important to realize that both the hyperfine coupling constant (a) and the nuclear Larmor frequencies (${\displaystyle \nu _{\mathrm {n} }}$) are determined when using the ENDOR method.^[6]

${\displaystyle \nu _{\mathrm {1} }=|\nu _{\mathrm {n} }-a/2|}$

${\displaystyle \nu _{\mathrm {2} }=|\nu _{\mathrm {n} }+a/2|}$

Requirement for ENDOR

One requirement for ENDOR is the partial saturation of both the EPR and the NMR transitions defined by

${\displaystyle \gamma _{e}^{2}B_{1}^{2}T_{1e}T_{2e}\geq {1}}$

and

${\displaystyle \gamma _{n}^{2}B_{2}^{2}T_{1n}T_{2n}\geq {1}}$ ^[5]

where ${\displaystyle \gamma _{\mathrm {e} }}$ and ${\displaystyle \gamma _{\mathrm {n} }}$ are the gyromagnetic ratio of the electron and the nucleus respectively. ${\displaystyle B_{1}}$ is the magnetic field of the observer which is microwave radiation while ${\displaystyle B_{2}}$ is the magnetic field of the pump which is radio frequency radiation. ${\displaystyle T_{\mathrm {1e} }}$ and ${\displaystyle T_{\mathrm {1n} }}$ are the spin-lattice relaxation time for the electron and the nucleus respectively. ${\displaystyle T_{\mathrm {2e} }}$ and ${\displaystyle T_{\mathrm {2n} }}$ are the spin-spin relaxation time for the electron and the nucleus respectively.

EI-EPR

ENDOR-induced EPR (EI-EPR) displays ENDOR transitions as a function of the magnetic field. While the magnetic field is swept through the EPR spectrum, the frequency follows the Zeeman frequency of the nucleus. The EI-EPR spectra can be collected in two ways: (1) difference spectra^[7] (2) frequency modulated rf field without Zeeman modulation.

This technique was established by Hyde^[7] and is especially useful for separating overlapping EPR signals which result from different radicals, molecular conformations or magnetic sites. EI-EPR spectra monitor changes in the amplitude of an ENDOR line of the paramagnetic sample, displayed as a function of the magnetic field. Because of this, the spectra corresponds to one species only.^[5]

Double ENDOR

Double electron-nuclear-double resonance (Double ENDOR) requires the application of two rf (RF1 and RF2) fields to the sample. The change in signal intensity of RF1 is observed while RF2 is swept through the spectrum.^[5] The two fields are perpendicularly oriented and are controlled by two tunable resonance circuits which can be adjusted independent of each other.^[8] In spin decoupling experiments,^[9] the amplitude of the decoupling field should be as large as possible. However, in multiple quantum transition studies, both rf fields should be maximized.

This technique was first introduced by Cook and Whiffen^[10] and was designed so that the relative signs of hf coupling constants in crystals as well as separating overlapping signals could be determined.

CP-ENDOR and PM-ENDOR

The CP-ENDOR technique makes use of circularly polarized rf fields. Two linearly polarized fields are generated by rf currents in two wires which are oriented parallel to the magnetic field. The wires are then connected into half loops which then cross at a 90 degree angle. This technique was developed by Schweiger and Gunthard so that the density of ENDOR lines in a paramagnetic spectrum could be simplified.^[11]

Polarization modulated ENDOR (PM-ENDOR) uses two perpendicular rf fields with similar phase control units to CP-ENDOR. However, a linearly polarized rf field which rotates in the xy-plane at a frequency less than the modulation frequency of the rf carrier is used.^[5]