Magnetic energy levels

The essence of NMR spectroscopy is to measure the separation of the magnetic energy levels of a nucleus. 

Bagguley, D.M.S. and Griffith, J.H.E. 1947. Paramagnetic resonance and magnetic energy levels in chrome alum. Nature 160 532-533. 

At equilibrium, the nuclear magnetic energy levels are populated according to a Boltzmann distribution which favors the lower state. For the two orientations relative to B0 of nuclei with 1 = the spin populations may be symbolized by N+ and N (Fig. 1.3 (b)). The distribution 7V+/7V can be expressed by the Boltzmann factor, recalling that... 

Relaxation can be described in terms of the magnetization vector components. At resonance, the equilibrium magnetization M0 parallel to B0 decreases to Mz, due to the transitions between the nuclear magnetic energy levels caused by the alternating field B,. Following resonance, the equilibrium of the nuclear spins with their lattice and with each other is restored by relaxation. 

The spins of two proximate nuclei A and X in a molecule undergo coupling, i.e. the signals of A and X are split, but only if the life-times of these nuclei in their magnetic energy levels (Fig. 1.3) are sufficiently large (scalar coupling). 

Having determined the magnetic energy levels eafiB) (as eigenvalues of the interaction Hamiltonian) we can proceed with the apparatus of the statistical thermodynamic by defining the (magnetic) partition function... 

The full list of parameters that enter the calculation of the magnetic energy levels is ... 

To the degree that the thermal motions of the neighboring nuclei in the lattice (the term is employed in dealing with both liquids and solids) have Fourier components at the precession frequency of the nuclei (for protons commonly 40 or 60 Mc/sec.) they will induce transitions between the magnetic energy levels and will shorten Tv In liquids composed of relatively small, compact molecules (but not in polymer... 

FIG. 12.5 The splitting of magnetic energy levels of protons, expressed as frequency vQ in varied magnetic field B0. 

The magnetic energy level is degenerate with a total of 27+ 1 quantum states. In the presence of an external magnetic field these quantum states have different energies, each with a quantum number m- For example, the proton for which 7= 1/2 has two energy levels with = 1/2 or —1/2, and the deuteron for which 7=1 has three energy levels with m= 1, 0, or —1. 

This formula for the magnetic susceptibility is useful when the magnetic energy levels Et = f(B) are simple analytic functions of the magnetic induction from which the first and second derivatives are easily derived. Normally the situation is more complex and one should follow the following steps. 

In the absence of the external magnetic field the Hzk term is the only perturbation that influences the magnetic energy levels (except in the case of S = 1 /2 for which this vanishes). Then the energy levels in the zero magnetic field are obtained by solving the secular equation... 

The most frequent treatment is that the components of the magnetic susceptibility are derived for the case of D 0 and E = 0 by utilising the van Vleck equation. For this purpose we need the van Vleck coefficients as they result from the approximate expansion of magnetic energy levels. Based on Table 8.22, the non-zero van Vleck coefficients are arranged in Table 8.23 notice that... 

Electron spin resonance (electron paramagnetic resonance) is a spectroscopic method that measures differences between magnetic energy levels. This is the principal difference with respect to magnetic susceptibility measurements which accounts for the Boltzmann occupation of all energy levels. The fundamental resonance condition is... 

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