CW-ENDOR

ENDOR (Electron Nuclear Double Resonance) is a double resonance technique that combines the high resolution and nuclear selectivity of an NMR experiment with the sensitivity of an EPR experiment. George Feher developed the ENDOR technique for his studies of phosphorus-doped silicon (Phys. Rev., 103, 834 (1956)).

The ENDOR experiment is performed by monitoring the EPR signal intensity while sweeping a RF (Radio Frequency) signal to drive the NMR transitions. Double resonance experiments such as ENDOR require more effort than simpler EPR or NMR experiments, but ENDOR rewards you for your extra effort by supplying more insight into the structure, dynamics, and electronic structure of your samples.

The first step in performing an ENDOR experiment is to acquire an EPR spectrum in order to determine the magnetic field values at which the EPR signals occur. With this information, we can set the magnetic field to the top of the EPR absorption signal. The magnetic field is kept fixed at that value, hence the name static field value.

The next step is to saturate the EPR signal by applying high power microwaves. At low power, the EPR signal grows with the square root of P where P is the microwave power. At higher powers, the signal grows less quickly with more power and may actually diminish in intensity. This phenomenon is called saturation and it results from the electron spin system being heated by the microwaves. In the ENDOR experiment, we shall use the RF to desaturate the EPR signal.

The final step is to sweep the RF while monitoring the EPR absorption signal. The RF signal originates from the DICE ENDOR sytem located in the Elexsys console. (See Figure 3.1.) The signal is too low (in most cases) to drive the NMR transitions sufficiently for ENDOR, so an RF power amplifier is used to supply more power. The high power RF signal is applied to the coil inside the cavity and the power exits and is terminated in a large water-cooled 50 Ω. load. Care has been taken to make the electronics as broad-band as possible to minimize spurious signals and to ensure that all your couplings can be observed.

When the RF drives an NMR transition, the saturated EPR signal desaturates and increases in intensity, resulting in an ENDOR signal. (See Figure 3.2.) In order to detect the ENDOR signal with greater sensitivity, the RF is frequency modulated by the signal channel in a manner analogous to field modulation in EPR. The output signal of the bridge is sent to the signal channel for demodulation. A first derivative spectrum results.

Much of the insight comes from interpretation of the various interactions that can be measured. An appropriate spin hamiltonian that describes many experimental results is as follows:

H = µ_B B₀ g S + S A I + I P I + µ_N B₀ g_n I

equation 3.1

The largest interaction in most EPR experiments, the electron Zeeman effect, is represented by the first term in the spin hamiltonian. The constants in the term are: µ_B, the Bohr magneton; B₀, the externally applied magnetic field; g, the g matrix; and S, the electronic spin operator. This interaction determines the magnetic field at which the EPR signal will be centered. The Zeeman effect can give us some useful insight into the electronic structure of the radical; however, it does not tell us much about the molecular structure, spin distribution, or dynamics of our sample.

A nucleus may also have a Zeeman interaction: this is represented by the last term in the spin hamiltonian. The constants in the term are: µ_N, the nuclear magneton; B₀, the externally applied magnetic field; g_n, the nuclear g matrix; and I, the nuclear spin operator. The nuclear Zeeman interaction can provide information regarding the identity of the nucleus. (i.e. Is it a proton or a phosphorus nucleus?) Unfortunately, the nuclear Zeeman interaction is usually impossible to measure in an EPR experiment. It is, however, possible to measure this quantity in an NMR or ENDOR experiment.

Fortunately, the unpaired electron, which gives us the EPR spectrum, is very sensitive to its local surroundings. The nuclei of the atoms in a radical or complex often have a magnetic moment, which produces a local magnetic field at the electron. In a reciprocal fashion, the electron also produces a local field at the nucleus. The interaction between the electron and the nuclei is called the hyperfine interaction and is represented by the second term in the spin hamiltonian. S and I are the electron and nuclear spin operators respectively and A is the hyperfine matrix. The hyperfine splittings give us a wealth of information about our sample such as the identity and number of atoms which make up a radical or complex, their distances from the unpaired electron as well as electron spin densities at the nuclei.

The third term represents the electric quadrupolar interaction of the nucleus. I is the nuclear spin operator and P is the quadrupolar interaction matrix. This interaction is sensitive to the electric field gradients at the nucleus and hence gives insight to the local symmetry of the nucleus. The nucleus must have a spin greater than 1/2 in order for the quadrupolar term to have any effect on the energy eigenvalues of the spin hamiltonian. It is difficult to measure in an EPR experiment, but can be measured in an NMR or ENDOR experiment.

The spin hamiltonian in Equation 3.1 includes terms describing the anisotropy of the interaction. For the purposes of simplicity and clarity, we shall now only discuss the isotropic case (i.e. solution spectra) with spin 1/2 nuclei. We shall also use the approximation of first order perturbation theory to calculate the energy eigenvalues. (Also known as the high field limit.)

It is educational when discussing ENDOR experiments to look at the energy splittings in a magnetic field as well as the transitions for the EPR and NMR signals. First order perturbation theory yields the following energy eigenvalues for the spin hamiltonian in Equation 3.1:

E_{MS, MI} = gµ_BB₀M_I + h a M_SM_I

equation 3.2

M_S and M_I are the magnetic spin quantum numbers of the electron and nuclei respectively. h is Planck’s constant and a is the isotropic hyperfine coupling constant (in units of MHz). Some of the terms can get unwieldly with the large number of constants and we also are measuring frequencies in an ENDOR experiment. Therefore, with the following substitutions,

ν_e = gµ_BB₀/h and ν_n = g_nµ_NB₀/h

equation 3.3

equation 3.2 then becomes

E_MS,MI/h = ν_eM_S - ν_nM_I + aM_sM_I

equation 3.4

Next, let us examine the NMR transitions which are labeled as NMR in Figure 2.3. The selection rules for NMR (and ENDOR) transitions are:

ΔM_S = 0,      ΔM_I = ± 1

equation 3.7

which results in two NMR transitions:

ν_NMR = |ν_n ± a / 2 |

equation 3.8

if (ν_n > a / 2, (i.e. ν_NMR > 0.) the two NMR signals are centered at ν_n with a splitting of a. In most experiments the resonant frequency lies in the radio frequency region. Unlike EPR, the NMR spectrum contains information regarding the nuclear Zeeman interaction (ν_n), which facilitates the identification of the nucleus. (The nuclear g values, g_n, are well cataloged.)

We cannot distinguish between negative and positive frequencies (hence the absolute value brackets in Equation 3.8) in an NMR experiment. If ν_n < a / 2, the NMR signals will then be centered at a / 2 with a splitting of 2 ν_n.

If there is a second nucleus interacting with the electron, each of the EPR signals is further split into a pair, resulting in four signals. For N spin 1/2 nuclei, we will generally observe 2^N EPR signals. (If the N nuclei have identical splittings, only 2N+1 lines are observed and their intensity distribution follows the familiar Pascal’s triangle progression. See Figure 3.6.) As the number of nuclei gets larger, the number of signals increases exponentially. Many times there are so many signals that they overlap and we only observe one broad signal. This phenomenon is called inhomogeneous broadening. (See Figure 3.5.) All the important information we had hoped to obtain from the hyperfine splittings appears to be lost.

The problem is that each electron is surrounded by a large number of nuclei and the electron is interacting with them all simultaneously. On the other hand if we consider the nuclei, each nucleus interacts with only one electron because there are many more nuclei than unpaired electrons. The electron, which has a spin of 1/2, will split the NMR lines in two. The interactions between the nuclei themselves are so small compared to the hyperfine interaction that for all practical purposes we can ignore them. If we measure the nuclear splittings directly by an NMR experiment, we can obtain a much simpler spectrum because for N spin 1/2 nuclei, we will generally observe 2^N NMR signals instead of the 2^N signals in an EPR spectrum. For N nuclei with identical hyperfine splittings, only 2 NMR signals appear. The information that appeared lost in the EPR experiment (the hyperfine splitting) can still be obtained from an NMR experiment because of the reduction in the number of lines.

An apt analogy might be a spokesperson (the electron) and a crowd (the nuclei). Our goal is to obtain information (the hyperfine splittings) from the individual nuclei. In an EPR experiment, we ask the electron for the nuclear information, but the electron cannot supply us this information easily. The electron cannot hear the message because the large crowd of nuclei are shouting the information to the electron all at the same time. The message is lost due to the crowd’s noise. In an ENDOR experiment, we ask each individual nucleus in the crowd to stand up one at a time and pass the information on to the electron. Now we can make sense out of the message.

Everything we have discussed so far is valid for inhomogeneously broadened EPR spectra. There is also a second class of broadening, homogeneous broadening. For this case, the broadening is not caused by unresolved hyperfine structure and ENDOR does not yield any resolution enhancement.

In summary the ENDOR and NMR experiments offer superior resolution of the hyperfine coupling constants than EPR. The nuclear g value, g_n is also measurable in NMR and ENDOR, facilitating the identification of the nucleus. The only information lost because of the fewer number of signals is the number of equivalent nuclei. If there is sufficient resolution in the EPR signal, the number of equivalent nuclei can be rapidly determined by the intensity pattern. (See Figure 3.6.)