Table of Contents

PNMR Technique

This page goes into some of the details of how the technique of Pulsed Nuclear Magnetic Resonance uses magnetic fields to manipulate and make measurements of the bulk magnetization created by protons in a sample. This discussion is specific to the implementation of PNMR in the TeachSpin apparatus.

PNMR probe

The relevant components of the PNMR probe are shown in Fig. 6a. The brass block currently held within the fixed magnet contains the relevant components, and is depicted in Fig. 6b.

(a) (b)
Figure 6: PNMR sample probe. (a) Schematic of functional portions of the probe. (b) Annotated diagram of probe in situ.

A sample vial that holds a small amount of material may be positioned inside two orthogonal coils of wire. The receiver coil wraps around the sample volume along the $x$-axis. This coil uses magnetic induction to detect time varying magnetic fields along the $x$-axis. Along the $y$-axis is a second coil in a Helmholtz configuration with the sample volume at its center. This Helmholtz coil will be used to create short bursts of RF oscillating magnetic fields in the sample.

Recall that when the sample is at equilibrium in an external magnetic field, the only bulk magnetization present is of constant magnitude and pointed along the $z$-axis. However, our detectors receiver coil responds only to time-varying magnetic fields along the $x$-axis, so that magnetizations along the $z$-axis, $M_z$, cannot be measured directly. 

Figure 7 : A net magnetization in the $xy$-plane (the plane of the page) will oscillate about the $z$-axis (pointed into the page) with frequency $\omega$. This will be observed in the receiver coil as an oscillating component along the $x$-axis.

Only precessing magnetizations in the $xy$-plane, $M_{xy}$, can be measured directly as shown in Figure 7. The current induced in the receiver coil will vary sinusoidally in time at the precession frequency with an amplitude proportional to the magnitude of $M_x$.   In order to measure a bulk magnetization using PMNR, the orientations of the nuclei in the sample need to be manipulated in a manner which causes the magnetization of interest to be rotated into the $xy$-plane where it will precess and induce a current in the receiver coil. The rotation of the bulk magnetization is accomplished by sending pulses of RF oscillating current to the Helmholtz coils shown in green in Fig. 6.

Rotation of bulk magnetization

As an illustration of how PNMR manipulates the orientations of the nuclei in the sample, consider the following experiment.

Assume that our glycerin sample is at equilibrium in an external magnetic field ${\bf B} = B_0\hat{\bf z}$. As shown in Fig. 3, the Boltzmann distribution of proton spin states results in a bulk magnetization along the $z$-axis, ${\bf M} = M_0\hat{\bf z}$; we wish to measure the magnitude, $M_0$. In order to do this, we need to rotate this magnetization into the $xy$-plane so that it will induce a current in our receiver coil. To rotate, we use the Helmholtz coils to generate a time-varying magnetic field in the $xy$-plane,

${\bf B}_{RF} = 2B_1\cos(\omega t)\hat{\bf x}$. (18)

which can be decomposed into two counter-rotating fields as shown in Fig. 9:

${\bf B}_{RF} = B_1\left[\cos(\omega t)\hat{\bf x} + \sin(\omega t)\hat{\bf y} \right] + B_1\left[\cos(\omega t)\hat{\bf x} - \sin(\omega t)\hat{\bf y} \right]$. (19)

(The second term corresponds to a field rotating opposite to the precession. It can be shown to have no practical effects on the spin system and thus will be ignored in this analysis.)

The protons will precess about the net magnetic field which now includes both the fixed field along the $z$-axis and the time varying field in the $xy$-plane:

${\bf B}_{tot} = {\bf B}_{RF} + {\bf B}_Z= B_1\left[\cos(\omega t)\hat{\bf x} + \sin(\omega t)\hat{\bf y} \right] + B_0\hat{\bf z}$. (20)

Under the right conditions, this additional field will cause the protons to precess in such a manner that their resultant bulk magnetization gets rotated into the $xy$-plane. Let's see how this is done.

The classical equation describing the time variation of a magnetic dipole moment in a magnetic field is

$\boldsymbol \mu \times {\bf B} = \dfrac{1}{\gamma}\dfrac{d\mu}{dt}$. (21)

The equivalent for a net magnetization $\bf M$ is

${\bf M} \times {\bf B} = \dfrac{1}{\gamma}\dfrac{d \bf{M}}{dt}$. (22)

The analysis of the magnetization in this complicated time-dependent magnetic field can best be carried out in a rotating coordinate system. The coordinate system of choice is rotating at the same angular velocity as the rotating magnetic field with its $z$-axis in the direction of the static magnetic field. In this rotating coordinate system, the rotating magnetic field appears to be stationary and aligned along the $x^*$-axis. (See Fig. 8.)

However, from the point of view of the rotating coordinate system, $B_0$ and $B_1$ are not the only magnetic fields. An effective field along the $z^*$-direction, of magnitude $-\omega/\gamma$ must also be included. Let us justify this new effective magnetic field with the following physical argument. Eq. (22) predicts the precession of a magnetization in a constant magnetic field. Suppose one observes this precession from a rotating coordinate system which rotates at the precession frequency. In this frame, the magnetization appears stationary and in some fixed position. The only way a magnetization can remain fixed in space is if there is no torque on it. If the magnetic field is zero in the reference frame, then the torque on $\bf{M}$ is always zero no matter in what direction  $\bf{M}$ is oriented. The magnetic field is zero (in the rotating frame) if we add the effective field ${\bf{B}}_{eff} = -\omega / \gamma \hat{\bf z}^*$ which is equal to $B_0 \hat{\bf z}^*$. Transforming the magnetic field expression in Eq. (20) into such a rotating coordinate system, the total magnetic field ${\bf B}^*_{tot}$ in the rotating frame,

$\mathbf{B}^*_{tot} = \mathbf{B}_1 \hat{x}^* + \left(\mathbf{B}_0 - \dfrac{\omega}{\gamma}\right)\hat{z}^*$, (23)

is shown in Fig. 10.

The classical equation of motion of the magnetization as observed in the rotating frame is then

${\bf M} \times {{\bf B}^*_{tot}} = \dfrac{1}{\gamma}\dfrac{d \bf{M}}{dt}$. (24)

Suppose now, we create a rotating magnetic field such that $\omega_0 = \gamma B_0$. In that case, ${\bf B}_{tot}^* = B_1 \hat{\bf i}^*$ is a constant magnetic field in the $x^*$- direction. Then, the magnetization  $M_z$ begins to precess about this magnetic field at a rate $\Omega = \gamma B_1$ (in the rotating frame). Thus, by applying the rotating field for a controlled duration (an RF pulse), it is possible to change the net magnetization through a controlled angle away from the $z$-axis. A $90^\circ$ pulse, for example, is an RF pulse of just sufficient duration to rotate the net magnetization $90^\circ$ away from the $z^*$-axis, i.e., into the $xy$-plane. If the rotating field is applied for twice this time, then the net magnetization will be rotated into the $-z^*$ direction. If the pulse is left on four times as long the magnetization will be back where it started, with $M_z$ along the $z^*$-axis. These RF pulses are labeled as follows:

In the laboratory reference frame the magnetization not only precesses about $B_1$ but also precesses about $\hat{\bf z}$ during the pulse. It is not possible, however, to observe the magnetization during the pulse. Pulsed NMR signals are observed after the RF pulse has been applied. Because of the orientation of the pick-up coil, wound around the sample vial, the measurable effect is the precession of the magnetization in the $xy$-plane.

Figure 8: In the rotating coordinate system, the rotating magnetic field ${\bf B}_{RF}$ appears to be stationary and aligned along the $x^*$-axis.
Figure 9: The oscillating field can be decomposed into two counter-rotating fields.
Figure 10: Precession shown in the rotating frame. (Source: [1])

Free induction decay (FID)

By applying a 90º pulse to a sample at equilibrium, the magnetization $M_z$ can be rotated into the $xy$-plane where it will precess about $B_0 \hat{\bf z}$. This precessing magnetization in the $xy$-plane, which we now call $M_{xy}$, induces a current in the receiver coil while relaxing back to equilibrium. The resulting signal from the receiver coil is called the free induction decay, or FID and is illustrated in Fig. 11. The peak amplitude of the FID is proportional to the magnitude of $M_z$ just before the application of the $90^\circ$ pulse.

Figure 11: A schematic of the expected free induction decay (FID) signal. Note that this signal corresponds to the detector out, which strips out high-frequency components.