Two Spin System

2.Two Spin System

Nuclear Overhauser Effect (NOE) is caused by relaxation of one nuclear spin by a neighbouring spin. To understand the nature of the NOE, we have to look at the two-spin system A and X. Since NOE does not involves coherences, but merely polarization, i.e. population differences between the α and β states, we can use the energy level diagram (Fig.1). In a system consisting of two NMR-active nuclei/protons of different species, A and X. Here we assume scalar coupling, J_AX = 0. (J is the magnitude of coupling constant. Since J arises from the spin of the proton, its magnitude is not dependent on the applied magnetic field.)

If the two are close enough together, leading to perturbation of the populations of A and hence a change in intensity, and thus an NOE at A arising from initial perturbation of X. This effect then allow us to detect or measure the short distance between A and X. The reason for this is that spontaneous relaxation is extremely slow. The main way in which a spin can undergo a transition from one spin state to another is if that transition is stimulated, usually by a change in the magnetic field strength at the nucleus, which must be happening at a rate that matches the frequency of the transition. In our simple two-spin system, there are four possible energy states and six possible transitions between them. Transitions between level N₁ and N₃, and between level N₂ and N₄ are transitions of the A-nucleus and the transitions between level N₁ and N₂, and between level N₃ and N₄ are those of the X-nucleus. These transitions are observable and allowed. The energy difference between the two states (i.e. the frequency of the transition) corresponds to the frequency of spin A and spin X  in case of transitions between level N₁ and N₃, and between level N₂ and N₄; transitions between level N₁ and N₂, and between level N₃ and N₄ respectively.

According to NOE experiment, saturate the transition of one nucleus (e.g. A). Signal intensities for X-nucleus would not change. System tries to restore the equilibrium by spin-lattice relaxation and predominantly via a dipolar mechanism. Only those transitions in which the magnetic quantum no., m changes by 1, i.e. single quantum transitions, are allowed. Also called as “flip-flop” transitions in which the two spins A and X exchange energy. The frequency of this transition is the difference in chemical shift between A and X., so for a homonuclear system it is about a few MHz.

  ∆m= ±1                

Fig.2 shows all the possible and theoretically allowed relaxation processes, with their transition probabilities W.

·Four probabilities W₁ correspond to the single quantum transitions i.e. only one of the two (either W_1A or W_1X .i.e. αα →βα and αβ→ββ or αα→αβ and βα→ββ respectively).

·Transitions N₄-N₁ and N₃-N₂ are new.

·W₂ transition denote the probability for relaxation of the spin system via double quantum transitions i.e. simultaneous spin flip of both spins in the same direction. (e.g. αα→ ββ or βα →αβ).

∆m = 2

·W transition denote the probability for relaxation of the spin system via zero quantum transitions i.e. simultaneous spin flip α→β for one spin and α→β for the other.

∆m = 0

·W₂ and W transitions cannot be excited by Electromagnetic radiation; they are spectroscopically forbidden transitions and therefore cannot be observed in the NMR spectrum.

However, both are allowed transitions in relaxation.

·W₂ and W transitions are determined almost entirely by dipole-dipole relaxation. For the dipolar relaxation transitions of one nuclei/proton (e.g.X) are simulated by molecular tumbling at higher frequency (e.g.500MHz for ¹H in an 11.7 T field). This mechanism is normally referred as dipolar relaxation, because it is relaxation of one nuclei by the dipole of the other nuclei (e.g. A). This transition has no effect on the intensity of A, so does not produce an NOE. Dipolar relaxation gets weaker as the distance between A and X increases, at a rate proportional to r^-6.