Spin-Spin Interactions
What are spin-spin interactions?
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In the previous Q&A we showed how molecular motion of two dipoles could create locally fluctuating magnetic fields resulting in T1 and/or T2 relaxation. A somewhat different way to look at the dipole-dipole interaction with a quantum mechanical bent involving energy levels is presented below. This analysis provides equivalent final results but slightly different insights.
Consider the system of two spins shown below, which might be the two hydrogen protons on a water molecule (HOH). Depending on the initial orientations of the spins, several different outcomes may result. Most of these outcomes result in energy exchange and hence both T1 and T2 relaxation; other outcomes do not result in a net change in energy and T2 relaxation only.
Consider the system of two spins shown below, which might be the two hydrogen protons on a water molecule (HOH). Depending on the initial orientations of the spins, several different outcomes may result. Most of these outcomes result in energy exchange and hence both T1 and T2 relaxation; other outcomes do not result in a net change in energy and T2 relaxation only.
In the diagram right I have shown two spins (nuclei/protons) in pure "spin-up" or "spin-down" states separated by energy quanta (ΔΕ = hfo). This has been done for simplicity/clarity only. As stated many times elsewhere on this site, spins are not restricted to these absolute orientations, but by doing so here the reasoning will hopefully be made easier to follow. |
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Let us first assume one nucleus comes into the interaction in the pure "spin-up" state and the other begins in the pure "spin-down" state. The interaction between the two may cause only a single nucleus to change state, resulting in either two "spin-up" nuclei or two "spin-down" nuclei. In either case, an energy transfer at the Larmor frequency will have occurred resulting in T1 (and therefore also T2) relaxation. From this new {up-up} or {down-down} condition, a double transition is now possible, with both spins simultaneously changing to the opposite paired state. Such a double transition also involves energy exchange and results in both T1 and T2 relaxation.
An interesting interaction is seen in the middle of the diagram below, where an initially {up-down} pair of spins interacts resulting where both flip resulting in a final {down-up} state for the system. This is known as a "flip-flop". Although one spin has gained energy and the other lost energy, no net change in total system energy has occurred and hence no T1 relaxation. However, the interaction has randomly altered the angular momentum of both spins, so they no longer have any constant phase relationship to each other or to other spins in the sample. Hence, T2 relaxation alone has taken place.
A spin-spin interaction where there is a simultaneous state change of both spins (either of the double transition or flip-flop variety) is called cross-relaxation.
An interesting interaction is seen in the middle of the diagram below, where an initially {up-down} pair of spins interacts resulting where both flip resulting in a final {down-up} state for the system. This is known as a "flip-flop". Although one spin has gained energy and the other lost energy, no net change in total system energy has occurred and hence no T1 relaxation. However, the interaction has randomly altered the angular momentum of both spins, so they no longer have any constant phase relationship to each other or to other spins in the sample. Hence, T2 relaxation alone has taken place.
A spin-spin interaction where there is a simultaneous state change of both spins (either of the double transition or flip-flop variety) is called cross-relaxation.
References
Levitt MH. Spin Dynamics. Basics of Nuclear Magnetic Resonance (2nd ed). Wiley, Chichester, 2012, pp 369-408.
Levitt MH. Spin Dynamics. Basics of Nuclear Magnetic Resonance (2nd ed). Wiley, Chichester, 2012, pp 369-408.