The Resonance Condition

Gyromagnetic Ratios and Resonance Frequencies for Some Common Nuclei at Different Magnetic Flux Densities (Bo) 

The absorption of energy stimulating transitions between the two levels of energy in a magnetic field is called magnetic resonance. Thus, the resonance condition obtained by equaling Eo to AE can be expressed either as  

---

In an electron spin resonance spectrometer, transitions between the two states are brought about by the application of the quantum of energy hv which is equal to g H. The resonance condition is defined when hv = g H and this is achieved experimentally by varying H keeping the frequency (v) constant. Esr spectroscopy is used extensively in chemistry in the identification and elucidation of structures of radicals. 

The locations of the maxima of the -field and the E-field are different depending on the mode chosen for the EPR experuuent. It is desirable to design the cavity in such a way that the B field is perpendicular to the external field B, as required by the nature of the resonance condition. Ideally, the sample is located at a position of maxuuum B, because below saturation the signal-to-noise ratio is proportional to Simultaneously, the sample should be placed at a position where the E-field is a minimum in order to minimize dielectric power losses which have a detrimental effect on the signal-to-noise ratio. 

With help of the four-level diagram of the =I= system (see figure BL15.8 two conniion ways for recording ELDOR spectra will be illnstrated. In freqnency-swept ELDOR the magnetic field is set at a value that satisfies the resonance condition for one of the two EPR transitions, e.g. 4<- 2, at the fixed observe klystron frequency, The pump klystron is then turned on and its frequency, is swept. When the pump... 

As an example, we mention the detection of iodine atoms in their P3/2 ground state with a 3 + 2 multiphoton ionization process at a laser wavelength of 474.3 run. Excited iodine atoms ( Pi/2) can also be detected selectively as the resonance condition is reached at a different laser wavelength of 477.7 run. As an example, figure B2.5.17 hows REMPI iodine atom detection after IR laser photolysis of CF I. This pump-probe experiment involves two, delayed, laser pulses, with a 200 ns IR photolysis pulse and a 10 ns probe pulse, which detects iodine atoms at different times during and after the photolysis pulse. This experiment illustrates a frindamental problem of product detection by multiphoton ionization with its high intensity, the short-wavelength probe laser radiation alone can photolyse the... 

The simplest place to start is with a hydrogen atom. The experimental ESR spectrum shows two lines separated by 1420.4 MHz (often reported as a magnetic induction, since transitions occur at the resonance condition hv = In... 

In order to compute the lifetime of a phonon of energy E, one averages the Golden Rule scattering rate (7ig A /pCj )tanh(p /2) with respect to n(e, A), subject to the resonance condition E = y/EA + A [8, 11, 93]. This yields two contributions to the decay rate ... 

Depending on how the secondary magnetic field is applied, there are two fundamentally different types of spectrometers, namely, continuous wave (CW) and pulse Fourier transform (PFT) spectrometers. The older continuous wave NMR spectrometers (the equivalent of dispersive spectrometry) were operated in one of two modes (i) fixed magnetic field strength and frequency (vi) sweeping of Bi irradiation or (ii) fixed irradiation frequency and variable field strength. In this way, when the resonance condition is reached for a particular type of nuclei (vi = vo), the energy is absorbed and... 

Next the question arises of the extent to which the resonance condition is influenced by changes in the electrolyte. From eqn. 2.22 it follows that the H.F. conductance approaches zero for a very small or very large value of k, so that by differentating l/f p with respect to k one finds a peak position at... 

Figure 15 illustrates the interplay between the time scales of intermediate state decay, final state decay, and dephasing in determining the photon energy dependence of 8s. The final state in Fig. 15 is chosen to satisfy the resonance condition when the intermediate state is resonantly excited, e/ e,- = 2(er — e,). 

ESR spectroscopy can be transformed into an imaging method, ESRI, for samples containing unpaired electron spins, if the spectra are measured in the presence of magnetic field gradients. In an ESRI experiment the microwave power is absorbed by the unpaired electrons located at point x when the resonance condition, Equation (10), is fulfilled. 

The term d(umn — co) is a function which requires that the resonance condition be satisfied that is, the microwave frequency must equal the resonance frequency. 

Often the resonance condition and the coupling constant are reported in terms of gauss rather than in energy units. In this case Eqs. (29B), (30B), and (31B) are divided through by gfi and... 

One may observe from Eq. (5C) how the resonance condition is a function of the off diagonal components of the g tensor. 

Before we develop the resonance conditions for systems with hyperhne and with zero-held interactions, we return to the electronic Zeeman term S B as an example interaction to discuss a hitherto ignored complexity that is key to the usefulness of EPR spectroscopy in (bio)chemistry, namely anisotropy the fact that all interactions... 

Let us rewrite the resonance condition of an S = 1/2 system subject to the Zeeman interaction only as... 

When the hyperfine interaction is much smaller than the Zeeman interaction ( much means approximately two orders of magnitude or more), as is usually the case in X-band, then the resonance condition is... 

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

In this chapter we continue our journey into the quantum mechanics of paramagnetic molecules, while increasing our focus on aspects of relevance to biological systems. For each and every system of whatever complexity and symmetry (or the lack of it) we can, in principle, write out the appropriate spin Hamiltonian and the associated (simple or compounded) spin wavefunctions. Subsequently, we can always deduce the full energy matrix, and we can numerically diagonalize this matrix to obtain the stable energy levels of the system (and therefore all the resonance conditions), and also the coefficients of the new basis set (linear combinations of the original spin wavefunctions), which in turn can be used to calculate the transition probability, and thus the EPR amplitude of all transitions. 

From the first two /22i s we obtain the resonance condition for the non-Kramer s doublet ... 

---