Pulsed NMR: Measurement of Relaxation Times

Nuclear Magnetic Resonance serves as an important technique in many fields, including medicine (where it is known as MRI), chemical spectroscopy, and even quantum computation research. At its most basic, NMR is about the coherent manipulation and observation of nuclear dipole moments. Our work here deals exclusively with the hydrogen nucleus, which, being merely one proton, is both the simplest example and one of the most common nuclei in NMR. The standard NMR configuration (Figure 1) places a macroscopic sample (in our case, a small vial of aqueous glycerol) in a strong magnetic field B0ẑ (for us, B0 ∼ 1750G). We begin in thermal equilibrium: the magnetic dipole moments ~ μ prefer to align with the magnetic field, but, at room temperature, the alignment energy is vastly outscaled by available thermal energy, so there is only a slight directionality. In a macroscopic sized sample, this will result in a detectable net magnetization along ẑ. Furthermore, as one can easily show, the magnetic dipole moments of the nuclei will precess with the Larmor frequency ω0 = γB0 around the magnetic field (at ∼7MHz for our parameters). But, since the dipole moments are in general out of phase, this results in no (initial) net magnetization in tranverse (x-y) plane. As discussed in [5], the sample magnetization can be manipulated by application of a small oscillating transverse magnetic field ~ B1 = B1 sin(ωt)x̂. If the driving frequency ω is near resonance with ω0 (typically, |ω − ω0| ∼10 kHz), then the pulse can coherently rotate the dipole moment along the polar angle, by an amount proportional to the duration of its application (typically tens of μs). Most commonly, we will choose pulses to acheive π or π/2 rotations, and observations of the oscillating magnetic moment are made along the x̂ direction. The relaxation times of the system determine how long one can mantain coherent control over the magnetization. The spin-lattice relaxation, T1, describes the timescale for thermal reequilibriation, the recovery of the system’s original longitudinal magnetization after a pulse. The spin-spin relaxation, T2, describes the timescale for transverse decoherence due to the interactions of neighboring spins; this coupling induces relative phase differences, reducing coherence and shrinking the net magnetization. However, in practice, one actually observes a transFIG. 1: Application of a resonant transverse B-field can rotate the magnetization.