[1] Pulsed NMR Apparatus Manual, Teach Spin Inc.

[2] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York, 1996.

[3] H. Carr and E. Purcell, "Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments", Phys. Rev. 94(3), 630 (1954).

[4] N. Bloembergen, E.M. Purcell, and R.V. Pound, "Nuclear Magnetic Resonance", Nature 160( 4066), 475 (1947).

[5] E. L. Hahn, "Spin Echoes", Physical Review 80(4), 580 (1950).

[6] N. Bloembergen, "Encounters in Magnetic Resonances", World Scientific Series in 20th Century Physics 15, World Scientific,1996.

[7] R. R. Ernst and W. A. Anderson, "Application of Fourier Transform Spectroscopy to Magnetic Resonance", The Review of Scientific Instruments 37(1), 1966.

Pulsed nuclear magnetic resonance (PNMR) is an experimental technique used to study the response of magnetic nuclei to an applied magnetic field. In this experiment you will learn the physics of how PNMR works and will make measurements of two characteristic relaxation time constants for protons in a mineral oil sample. These two time constants – and the techniques used to measure them – form the basis of medical MRI imaging.

Specifically, your objectives for this experiment include the following:

• to learn to use the Pulse Programmer apparatus and the large, water-cooled electromagnet;
• to establish a resonance in samples of mineral oil and teflon, and to use these measure the gyromagnetic ratio of the proton and of fluorine nuclei, respectively;
• to measure the $T_1$ relaxation time in mineral oil using the inversion-recovery method; and
• to measure the $T_2$ relaxation time in mineral oil using the Hahn spin-echo method.

Here we wish to examine the effect of magnetic fields on protons and other particles having magnetic moment and angular momentum. It is noteworthy that the underlying physics of nuclear magnetic resonance is very similar to that of electron spin resonance and optical pumping.

We will first look at the behavior of a spinning magnetic dipole moment in an external magnetic field, then we will extrapolate that behavior to a large ensemble of protons.

It is useful to model nuclei as spinning bar magnets with an intrinsic magnetic dipole moment, $\boldsymbol{\mu}$, and an angular momentum, $\bf L$. In free space, a magnetic moment, $\boldsymbol{\mu}$, is free to point in any direction. However, if an external magnetic field $\bf B$ is present, $\boldsymbol{\mu}$ will try to align itself with the external field. When we consider the effect of conservation of angular momentum, we find that $\boldsymbol{\mu}$ will not fully align with $\bf B$, but will instead precess about the axis defined by $\bf B$. This behavior is analogous to a spinning top precessing in a gravitational field. In a material containing large numbers of such nuclei, the sum of all the aligned nuclear dipole moments results in a net (bulk) magnetization in the sample. It is the magnitude of these bulk magnetizations, and their behavior over time which is measured in PNMR. Of particular interest are the characteristic times associated with the following:

1. How long does it take a randomly oriented ensemble of magnetic nuclei to become aligned? This time scale is $\mathrm{T}_1$, known as the spin-lattice relaxation time
2. How long does it take for nuclei precessing in phase to get completely out of phase due to nearest neighbor interactions? This is $\mathrm{T}_2$, known as the spin-spin relaxation time

The rest of the theory section will focus specifically on protons (hydrogen nuclei), though it is generally applicable to any magnetic nuclei.

Consider the behavior of a bar magnet with dipole moment $\boldsymbol{\mu}$ in a magnetic field $\bf B$ as shown in Fig. 1. When placed in an external magnetic field $\bf B$, the dipole will feel a torque given by

that will act to align $\boldsymbol{\mu}$ with $\bf B$ and minimize the energy

Figure 1: Spinning bar magnet in an external magnetic field.

However, if we add angular momentum $\bf L$ to the bar magnet, the torque will not align the moment with the field, but will instead causes it to precess about the axis defined by $\bf B$. This is analogous to a spinning top; gravity acts to pull the top down to the table, but the angular momentum keeps the top spinning at a constant angle with respect the vertical. The frequency at which the bar magnet precesses is given by

where we introduce the gyromagnetic ratio,

For a bar magnet, the magnetic moment and angular momentum can be separately adjusted, but in the case of elementary particles, the ratio $\gamma$ is intrinsic and fixed. Note that there is a component of $\boldsymbol{\mu}$ which projects onto the $z$-axis, $\mu_z$, and which does not vary as the dipole precesses about the z-axis. Additionally, there is a component projecting into the $xy$-plane,$\mu_{xy}$, which does precess about the $z$-axis with an angular velocity $\omega$. The vector components of $\boldsymbol{\mu}$ can thus be written as

Like the spinning bar magnet described above, protons have a magnetic dipole moment and carry angular momentum. In the proton's case, the moment $\boldsymbol{\mu}$ is parallel and points in the same direction as the angular momentum,$\bf I$. (In the nuclear case, the angular momentum is due to intrinsic nuclear spin and is designated by $\bf I$ rather than $\bf L$ to distinguish it from other types of angular momentum, such as orbital angular momentum). When placed in an external magnetic field, ${\bf B} = B_0 \hat{\bf z}$, the proton will precess about that field. The most significant difference from the bar magnet, though, is that the angular momentum $\bf I$ is quantized along the $z$-axis (which is defined to be the axis of the external magnetic field) such that  ${\bf{I}}_Z = m_l\hbar$, where $m_l$ is the so-called magnetic quantum number. This number obeys $-s < m_l < s$ (in integer steps) where $s$ is the intrinsic spin quantum number. For protons with $s = 1/2$, we have $m_l = \pm 1/2$. Physically, this means that a proton can have the component of its angular momentum either aligned preferentially in the direction of $\bf B$ ($m_l = + 1/2$) or in the opposite direction ($m_l = - 1/2$). Using Eqs. (2) and (4) above, and substituting the nuclear spin $\bf I$ for $\bf L$, we can write the energy of the proton's spin state as

where again ${\bf{I}} _Z = m_l\hbar$. For $m_l = \pm 1/2$ , the two possible energy states for a proton in an external magnetic field are shown in Fig. 2. Note that because of the negative sign, the orientation where the magnetic moment is preferentially aligned with the field $m_l = + 1/2$) is a lower energy state than the case where the moment is anti-aligned ($m_l = - 1/2$).

Figure 2: The two allowed energy states of a proton (spin 1/2 particle) in a magnetic field.

The energy separation$\Delta E$ between the two states can be written in terms of an angular frequency,

where we use the same relation between angular frequency, gyromagnetic ratio, and field as the classical bar magnet, Eq. (3):

For the proton, the resonant frequency, $f_0 = \omega_0 / 2\pi$, is

and its gyromagnetic ratio $\gamma$ is

NOTE: Gauss (symbol: G) is the traditional unit for magnetic fields in nuclear magnetic resonance, but Tesla (symbol: T) is the SI unit, where 1 T = $10^4$ G.

In PNMR, one does not directly observe the behavior of individual nuclei (protons in our case). What is measured are the macroscopic bulk magnetization, $\bf M$, arising from the alignment of large numbers of nuclei.

First consider consider the case of a large number (order $10^{19}$) of protons when no external magnetic field is present. In this case, there is no preferred direction in space, no quantization axis, and no magnetic torque exerted on the protons. Thus, individual protons will be randomly oriented in space, there will be no precession, and the two energy states shown in Fig. 2 will be degenerate such that $\Delta E =0$. Therefore, if one were to do the vector sum of all of the proton dipole moments along any axis, the average would be zero; there would be no bulk magnetization in the sample.

If we now turn on an external magnetic field,

two important changes will occur to the ensemble of protons. One is that the protons will precess about the z-axis, which is now defined by the direction of the external magnetic field, at a frequency given by Eq. (8). The other is the spin energy states are no longer degenerate. At any given moment, some number of protons will be aligned parallel ($m_l = +1/2$) to ${\bf B}$ and the remainder anti-parallel ($m_l = - 1/2$) to ${\bf B}$. If the sample is in thermal equilibrium, the average number of protons in each of the two possible states will be given by a Boltzmann distribution. If $N_1$ and $N_2$ are the number of spins per unit volume in the higher and lower energy states respectively, then the population ratio $(N_1/N_2)$ in thermal equilibrium is,

where $T$ is the absolute temperature, $k_B$ is the Boltzmann constant, and $\Delta E = \hbar\omega_0$ is the energy difference between the two states. At room temperature, $k_BT \approx 1/40 \,eV$ and the ratio between $N_1$ and $N_2$ is nearly equal to, but not quite, 1; there is a slightly larger number of protons in the low energy state (i.e. aligned along +$z$) than in the high energy state. Now, when we sum up all of the proton dipole moments,

we get a non-zero net magnetization along the $z$-axis,

Because the angular momentum is quantized only along the $z$-axis, there is no preferred direction in the $xy$-plane. Therefore, when the precessing components of the proton dipole moments in the $xy$-plane are summed, ($\mu_{xy}$, see Fig. 1), they average out to zero. Figure 3 depicts this equilibrium bulk magnetization.

Figure 3: Bulk magnetization resulting from an ensemble of protons at equilibrium in an external magnetic field.

Note that individual protons in the sample are continually flipping back and forth between the two possible orientations (i.e. energy states) with respect to the $z$-axis. A proton in the low energy state can be excited to the upper state by absorbing a quanta of energy,$\Delta E = \hbar \omega_0$, from the local environment. Similarly, a proton in the excited state can de-excite to the low energy state by giving up a quanta of energy. It is the physical processes by which the protons in the sample exchange energy with their environment (referred to as the lattice) that determine how frequently proton spins flip between energy states. How long it takes an ensemble of protons to reach equilibrium depends on how rapidly they are changing states which in turn depends on the details of their environment. In liquids, the protons and other molecules are loosely bound and undergo Brownian motion which results in random fluctuations in the local electric and magnetic fields the protons see. These fluctuating fields are an important factor in driving the rate of spin flips in liquids.

In solids, the protons are locked into a rigid lattice structure where quantized lattice vibrations and the exchange of phonons come into play. The more rapidly protons flip spin states, the faster the ensemble reaches equilibrium.

PNMR is typically used to make measurements of the characteristic time constants associated with how an ensemble of protons reaches equilibrium in an external magnetic field. These characteristic times are  $T_1$ and $T_2$. PNMR techniques – as we will see in more detail in the next section – operate roughly as follows:

1. Create a known, non-equilibrium configuration of spin states.
2. Measure the bulk magnetization either along the $z$-axis or in the $xy$-plane as a function of time as it relaxes back to equilibrium.

The time constant $T_1$ is a measure of how fast the magnetization on the z-axis ($M_Z$) relaxes to equilibrium. Recall that $M_Z$ arises from the fact that each proton has to be in one of two possible states, either parallel or anti-parallel to  $\bf B$. If there are more protons in one of the two states, the difference will add up to a net magnetization along the z-axis. Now imagine that our sample of protons is at thermal equilibrium; this yields the situation depicted in Fig. 4(a). We then do the following:

• At $t = 0$ we invert the Boltzmann distribution so that there are more particles in the high energy state than the low energy state. $M_Z$ has the same magnitude, but now points along the negative z-axis as shown in Figure 4(b). This is a non-equilibrium state. Immediately after inverting the Boltzmann distribution, the system begins relaxing back to equilibrium. Since the rate of spin flips from high to low energy is greater than the reverse process, over time the high energy state becomes less populated while the low energy state becomes more populated.
• We allow the system to relax for a small amount of time (small relative to the time it takes the sample to relax all the way back to equilibrium) and then make a measurement of $M_Z$. We see that $M_Z$ has relaxed partway back to the equilibrium state. (See Fig. 4(c).)
• We wait some more time, and at time $t2$ make another measurement of $M_Z$ which has continued to relax back to equilibrium. (See Fig. 4(d).)
• We continue this process until $M_Z$ has fully relaxed back to the equilibrium state. (See Figs. 4(e) and 4(f).)

The rate at which the system approaches equilibrium is proportional to how far away from equilibrium it is ($M_0$ - $M_Z$) and inversely proportional to the characteristic relaxation time ($T_1$):

Integrating this with the initial condition $M_Z (t=0) = -M_0$, we get

Figure 4: Illustration of spin-lattice relaxation. The red arrows represent the bulk magnetization in the sample at various times in the process. The accompanying plots show graphically how the magnetization exponentially relaxes back to the equilibrium value with time.

The other time constant of interest is $T_2$, known as the spin-spin relaxation time. $T_2$ is a measure of how long it takes for a non-equilibrium bulk magnetization in the $xy$-plane to relax back to the equilibrium value of zero. Recall that when our ensemble of protons is in the presence of an external magnetic field and at equilibrium, all of the proton dipole moments will be precessing about the $z$-axis at the same frequency. However, since angular momentum is not quantized in the $xy$-plane, the precessing component of the proton dipoles, $\mu_{xy}$, are randomly oriented and consequently sum to zero. Therefore, there is no bulk magnetization in the $xy$-plane at equilibrium.   If, however, some or all of the dipoles are precessing in phase, summing over all $\mu_{xy}$ would result in a bulk magnetization in the $xy$-plane, $M_{xy}$. Furthermore, $M_{xy}$ would itself precess about the $z$-axis at frequency $\omega_0$. We normally assume that each proton dipole sees exactly the same magnetic field and therefore precesses at exactly the same frequency. If this were the case, the net magnetization$M_{xy}$ would keep the same magnitude and would maintain the constant precession indefinitely. That is not the case, though. Even if the applied magnetic field is perfectly uniform, each proton will still experience a slightly different net field because of nearest-neighbor effects. For example, each proton sees not just the external magnetic field, but also the magnetic dipole fields of the other protons (or other magnetic nuclei) in the sample which warp the local value. In addition, in a liquid sample, the nuclei are all moving about randomly, meaning that the local magnetic field fluctuations also change in time as particles come and go.

The net result is that there is a small spread of precession frequencies among the proton magnetic dipoles in the sample. The protons which initially are precessing in phase will gradually become more and more out of phase with each other until the entire ensemble is back to being randomly oriented. How quickly this happens is related to the magnitude of the spread of precession frequencies which is related to the magnitude of the local magnetic field fluctuations. This process is called spin-spin relaxation because it is driven by the interaction of the spin (equivalently, the dipole moment) of the proton with the spin of its neighboring protons.

As we did above when discussing $T_1$, let's start by imagining that we have a sample of protons at thermal equilibrium such that the net magnetization lies along the $z$-axis as in Fig. 4(a) above. We then do the following:

• At time $t_0 = 0$, we manipulate the protons in the sample such that the net magnetization (which was aligned with the $z$-axis, $M_z$), now points in some direction purely in the $xy$-plane, $M_{xy}$. In such a state, a significant number of the dipole moments are pointing in the same direction and precessing about the $z$-axis in phase at the frequency $\omega_0$ as shown in Fig. 5(a).
• Because the protons in the sample each feel a slightly different magnetic field due to the spin-spin interactions, their magnetic dipoles precess about the $z$-axis at slightly different frequencies. The $xy$-components of the proton spins will not remain in phase; proton dipoles precessing faster than $\omega_0$ get ahead of the pack while proton dipoles precessing slower than $\omega_0$ fall behind. Over time they will become increasingly out of phase as depicted in Figs. 5(b) through 5(d).

The differential equations which describe this relaxation are

and

the solutions of which are

and

Figure 5: Relaxation of bulk magnetization in the $xy$-plane. Note that in figure 5(a) the thick red arrow depicts the vector sum of all of the $xy$-components of the proton dipole moments as they precess in phase about the $z$-axis. In figures 5(b) through 5(d) the single single thick red arrow is replaced by a thinner red arrows. These thinner arrows now represent individual proton dipole components precessing in the $xy$ plane. The time evolution of the vector sum of the precessing proton dipole moments is shown in the accompanying plots to the right.

The technique of PNMR can be described in general as follows:

1. A sample containing magnetic nuclei is placed in an external magnetic field and allowed to come to equilibrium.

• In our case we will use mineral oil as a sample which provides a large number of protons as our magnetic nuclei.

2. Pulses of radio frequency (RF) oscillating magnetic field are used to reorientate the ensemble of proton spins into some non-equilibrium state.

• For example, we can apply the RF field for a finite time causing an inversion of the Boltzmann distribution as a precursor to measuring the $T_1$ time constant.
• Or, we can rotate the equilibrium bulk magnetization from the $z$-axis into the $xy$-plane where it will precess about the $z$-axis while decaying back to zero for measuring $T_2$.

3. Measurements of the bulk magnetization are performed as it relaxes back to equilibrium.

We will now discuss how our apparatus accomplishes this.

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.

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

Assume that our mineral oil 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,

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

(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:

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

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

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,

is shown in Fig. 10.

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

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:

• $90^\circ$ or $\pi/2$ pulse: $M_z \rightarrow M_y$
• $180^\circ$ or $\pi$ pulse: $M_z \rightarrow -M_z$
• $360^\circ$ or $2\pi$ pulse: $M_z \rightarrow M_z$

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])

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.

Figure 12 shows the components of the apparatus. We will summarize the role of each component here. For more detail on the operation of the electronics, refer to pages 14-25 of the TeachSpin manual in the laboratory.

Figure 12: Components of the PNMR experiment.

[A] Electro-magnet:

The electro-magnetic (Fig. 13) is capable of generating fields up to 10 T between the pole pieces. The strength of the field is determined by the amount of current flowing through the water-cooled coils and magnet power supply. Before turning on the electro-magnet power supply (Fig. 14), you must first turn on the water supply in the next room. You should see a small, steady stream emptying into the sink.

Figure 13: Water-cooled electromagnet

[B] Electro-magnet power supply:

Controls the current flowing through the coils of the electro-magnet, shown in Fig. 14. Only the coarse and fine current controls are used, do not change the settings of the other controls. Before turning on the power supply make sure that both the coarse and fine current controls are set to 0. When turning the power supply off at the end of the day, also make sure that the both current controls are set to zero. Turning the power supply on or off with the current controls not at zero can potentially damage the equipment.

Figure 14: Electro-magnet controls. Current controls are outlined in light blue.

[C] Sample Probe:

The sample probe is a rectangular brass box which slides snugly between the pole pieces of the electro-magnet. The interior of the probe is shown in Fig. 15. There is a hole in the top which accepts a sample vial containing a small amount of material to be studied. When inserted to the proper depth, the sample volume will be in the center of the receiver and Helmholtz coils. The receiver coil is used to detect time varying magnetic fields along the vertical axis (x-axis). The Helmholtz coils are used to create time varying magnetic fields along the horizontal axis (y-axis) which will be used to rotate the nuclei in the sample.

Mineral oil sample	Teflon sample

[D] TeachSpin Electronics Rack

The electronics rack houses three different modules. The power switch is located on the right hand side of the back of the rack. The three modules are as follows:

15 MHz Receiver

A low noise, high gain amplifier connected to the receiver coil in the sample probe. (See Fig. 16a.)

15 MHz Oscillator/Amplifier/Mixer

Sends pulses of 15 MHz ac current to the Helmholtz coils in the sample probe. (See Fig. 16b.) When these pulses turn on, and their duration, is determined by signals from the pulse programmer. Also contains a mixer which is used to compare the frequency of the oscillator signal with the signal induced in the receiver coil.

Pulse Programmer

Allows the user to set up sequences of pulses from the 15 MHz Oscillator. (See Fig. 16c.)

(a)	(b)	(c)
Figure 16: Teachspin electronics rack. (a) The receiver module, which connects to the detector probe and processes the signal. (b) The pulse programmer, which controls the pulses that are sent to the Helmholtz coils. (c) The oscillator, amplifier, and mixer module, which sets the Helmholtz coil's frequency and detects the difference between the applied frequency and the detected frequency (Mixer out)

Make sure the controls of the pulse programmer are set as follows:

• A-width: halfway (This controls how long the applied pulse is)
• Mode: Int (This sets the pulse programmer to use the Internal function generator for timing)
• Repetition time: 100 ms and 10% (pulses should be produced 100 ms * 10% = 10 ms apart)
• Sync: A (sets the Sync out to trigger when pulse A starts.)
• A: On
• B: Off
• Sync Out: connected to oscilloscope external trigger input.

Insert the mineral oil sample. Connect the Detector Out from the 15 MHz receiver to channel 1 of the scope.  his output produces a signal whose amplitude is proportional to the magnitude of the emf in the detector coil, but with the high frequency (~15 MHz) oscillations filtered out. Make sure that the trigger point is centered on the scope display and the trigger is set to External. Set the time base of the scope to about 1 ms.

Figure 18: An example of the difference between the Detector Out (Ch 1) versus the RF Out (Ch 2). While it cannot be seen on this scale, the RF out signal has a ~15 MHz component that makes it appear quite noisy.

Now watch the signal on channel 1 as you slowly turn up the field on the electromagnet. You should find resonance (a peak of several volts) somewhere around a setting of 100 on the coarse current control of the electromagnet power supply. The signal you are looking at is the free induction decay (FID) signal. Over how wide of a range of coarse current settings can you observe resonance?  

The signal detected by the receiver coil will pick up the resonant frequency at which the proton magnetic moments precess. However, since this frequency is in the MHz range, it is difficult to make a precise measurement with just the oscilloscope. Instead, we will use a device called a frequency mixer, which takes two frequencies as input and produces a signal whose frequency is either the sum or difference of the two input frequencies. If we compare our measured resonance frequency to an internal reference frequency that we know well, we can tell how close we are to the reference frequency by looking at the beat frequency produced by the mixer output.

In our case, we can easily distinguish beats in the kHz range as we get close to matching the resonance and reference frequencies. (See Fig. 19.)

(A)	(B)	(C)
Figure 19: Examples of different mixer outputs: (A) Many beats, far from resonance (B) Some visible beats, still not at resonance (C) Three visible beats, nearly at resonance. Note that the scale on all three images are the same, but that (B) and (C) include the detector output signal.

Adjust either the magnetic field strength or the reference RF frequency until the beat frequency drops (close) to zero. At this point, we know that the protons are precessing at a frequency equal to the reference frequency.

NOTEBOOK:

Record the settings for the B-field and the RF frequency where you observe the strongest resonance.
Given the orientation of the detector coil in the sample probe, how can we tell empirically when the A pulse is a 90º pulse?  

Vary the A pulse width over its full range and observe the effect on the amplitude of the FID. Adjust the gain on the Receiver to nearly maximize the FID amplitude, with no clipping of the FID.

NOTEBOOK:

Describe the FID characteristics for a 90º, a 180º and a 270º A pulse.

Adjust the receiver gain so the FID amplitude is about 10 to 10.5 V (with no clipping) and the mixer output shows zero beat frequency. Experiment with the height of the sample in its holder. Iterate as needed to find the strongest signal.

No magnet produces a perfectly uniform field. However, it is possible to find a region of maximum uniformity (the sweet spot). To do so, move the sample probe around in the magnet gap while observing the changing shape of the FID. A uniform field is indicated by a long, smooth exponential decay of the FID. In a perfectly uniform field the exponential decay constant of the FID would be $T_2$, the spin-spin relaxation time constant, and would be on the order of 10 ms. Field inhomogeneities can dominate the relaxation time with a time constant of order 0.1 ms, and thus we cannot reliably measure $T_2$ with this method. 

NOTEBOOK and ANALYSIS:

The oscilloscope is capable of saving data to a flash drive. After inserting a device in the front USB port, use the save button near the top-right corner of the screen to send a screenshot and a .csv file to your flash drive.  You may want to use the “Run/Stop” button to find a suitable image to save, otherwise you get whatever is on the screen when you save.  Also note that the image saved is identical to the screen, so you may want to enable channel 1 or 2 individually to save separate images.

While at resonance (zero-beat condition) remove the sample tube containing the mineral oil sample and measure the magnetic field using the Hall effect gaussmeter. Calibrate your gaussmeter using the calibration magnets provided (see Fig. 20). Place the tip of the gaussmeter probe in the same region of the magnet where the sample sat. Note that the flat face of the Hall effect crystal must be perpendicular to the direction of the magnetic field being measured. Make a careful estimate of the uncertainty of this measurement.

NOTEBOOK and ANALYSIS:

From your measured values of resonant frequency and magnetic field, calculate γ for protons. 

(A)	(B)
Figure 20. (A) DC Gaussmeter (B) calibration magnet set

Repeat the gyromagnetic ratio measurement for Sample #2 (Fig. 15c) which is Teflon containing fluorine atoms. As fluorine atoms have a lower gyromagnetic ratio than protons, you will have to readjust the magnetic field (slowly!) to find the resonance.   

From your measured values of resonant frequency and magnetic field, calculate $\gamma$ for fluorine nuclei.
Discuss the sources of error between your values of the gyromagnetic ratios and the literature values, $\gamma_p = 2.675 \times 10^4$ rad/(G s) and $\gamma_F = 2.518 \times 10^4$ rad/(G s), respectively. 

We now know how to do the following:

• measure the magnitude of a bulk magnetization in the $xy$-plane which precesses about the $z$-axis, and
• reorient the bulk magnetizations in the sample.

The inversion recovery method uses a series of RF pulse pairs to measure the rate at which the magnetization along the $z$-axis relaxes to equilibrium. The method is as follows:

• Allow the sample to come to equilibrium in $B_0 \hat{\bf z}$.
• Apply a $180^\circ$ pulse to rotate the bulk magnetization from the +$z$-axis to the -$z$-axis. This is equivalent to inverting the Boltzmann distribution of proton spin states.
• After the $180^\circ$ pulse, the sample is allowed to relax back to equilibrium for  $\tau$ seconds.
• A $90^\circ$ pulse is applied to rotate the magnetization on the $z$-axis, which will have relaxed part way back to equilibrium, into the $xy$-plane where it will begin to precess and induce a current in the receiver coil. The amplitude of this FID signal is proportional to the magnitude of the magnetization which existed on the z-axis immediately before the 90º pulse was applied.

These steps are repeated, for different time intervals $\tau$. Using Eq. (15) (reproduced below), the time constant $T_1$ can be found by plotting FID amplitude versus delay time, $\tau$:

Procedure:

1. Establish resonance with the A pulse and set the width to 90º. Adjust (if necessary) so that the sample is again in the sweet spot of the magnet and set the gain so that the voltage is between 10 V and 10.5 V.
2. Set the A pulse to $180^\circ$, the first minimum.
3. Turn off the A pulse, turn on the B pulse and trigger the scope on the B pulse*.
4. Set the B width to give a $90^\circ$ pulse by maximizing the first FID signal.
5. Turn both A and B pulses on and set the scope to trigger on the B pulse.
6. Optimize the output signal by checking the magnet sweet spot and small tuning of the A and B pulse widths.
7. Adjust the delay time settings.  What effect does this have on your output signal?

NOTEBOOK and ANALYSIS:  > Measure  $M_z(t)$ (i.e. the amplitude of the FID) for enough different pulse lengths $\tau$ to observe the exponential decay of the magnetization. Make sure you have measured $M_0$, the asymptotic value of $M_z$. You can measure the relative amplitudes of $M_z(t)$ with the scope's cursor if the waveform is not changing its width. You can determine $T_1$ by fitting your data to the functional form of Eq. (15). Remember that our method can only measure the magnitude of $M_Z$ not the sign! What do you expect to measure when E $e^{-t/T_1} = 1/2$?

In principle $T_2$ can be extracted from the decay of the FID following a $90^\circ$ pulse. In reality, however, the situation is not so easy because protons in the sample experience different net magnetic fields due to two effects. One effect is the nearest neighbor spin-spin interaction which is a characteristic of the chemical environment that the protons are in. This is the quantity of physical interest. The second effect is the inhomogeneity of the applied magnetic field. No matter how well constructed, no real magnetic will produce a perfectly uniform magnetic field between its poles. Protons in different regions of the electromagnetic field experience different magnetic field strengths leading to different precession frequencies. As you should know by now, nature likes to arrange things so as to make it difficult for physicists to make measurements. So, it should not be surprising that the systematic effect of the electromagnet field inhomogeneity on the spread of precession frequencies in the sample dominates over the effect of the spin-spin interactions. Therefore, simply measuring the decay constant of the FID does not give $T_2$, but instead provides a measure of the field gradient of the electromagnet. In order to measure $T_2$ we need to use a very clever, but subtle technique developed by Erwin Hahn known as spin echo. Here we explain the logic of the spin echo measurement.

The spin echo method proceeds as follows:

• The sample is allowed to come to thermal equilibrium.
• A $90^\circ$ is applied to the sample to rotate $M_z$ into the $xy$-plane where it begins precessing and decaying due to the effects of both the spin-spin interactions and the field inhomogeneities.
• The sample is allowed to relax partway back to equilibrium for $\tau$ seconds. During this time protons which are in a stronger region of the electromagnet field will precess ahead of protons which happen to be in a weaker field region.
• A $180^\circ$ pulse is applied to the sample. This pulse causes any remaining magnetization in the $xy$-plane, $M_{xy}$, to precess $180^\circ$ about the $y^*$ axis (in the rotating frame of the proton). 
• After the $180^\circ$ $M_{xy}$ remains in the $xy$-plane but now the protons in the stronger field regions, which were getting ahead of protons in the weaker field regions, are actually behind the slower precessing protons in the weaker field regions.
• After  $\tau$ seconds, all of the proton dipole moments which  diverged  for time $\tau$ before the $180^\circ$ pulse will have reconverged. This reconvergence induces a current in the receiver coil which is called the echo pulse.

So what does the above sequence accomplish in terms of separating $T_2$ from the effects of the electromagnet? The key is that the 5th and 6th bullet points above only apply to precession frequency differences caused by the electromagnet, which are assumed to be constant over the time it takes to make the measurement. Thus, protons in a strong region of the electromagnet field which were precessing ahead of protons in weaker regions before the $180^\circ$ pulse, remain in that same strong region after the 180º pulse and as a result catch up to the slower precessing protons at the same rate as before the 180º pulse. This means that the effect of the electromagnet field inhomogeneities is reversible. This assumption however is NOT true for the nearest neighbor spin-spin interactions which fluctuate randomly on time scales small compared to the duration of a single measurement. So, while the $180^\circ$ pulse has the effect of reversing the magnetization loss due to the electromagnet, it has no effect on the loss of magnetization due to spin-spin interaction. Thus the amplitude of the echo is proportional to  $M_{xy}$ after it has been allowed to relax for $2\tau$ seconds under the effects of the spin-spin interaction only. Figure 21 illustrates the spin echo signal.

Figure 21: Expected signal from a spin-echo measurement.

By plotting the amplitude of the echo as a function of time $(t=2\tau)$, the time constant $T_2$ can be found from

The spin echo method due to Hahn uses a pulse sequence $90^\circ \rightarrow \tau \rightarrow 180^\circ \rightarrow \tau \rightarrow \textrm{"echo"}$ . Our starting point is the setup we have for the first FID experiment, with a $90^\circ$ A pulse, which we will use together with a $180^\circ$ B pulse. The $90^\circ$ pulse rotates the net magnetization away from the $z$-axis into the $x$-$y$ plane where it precesses at the Larmor frequency. We allow the magnetization to decay for a time $\tau$. The magnetization decays due to the differences in precession frequency caused by both the reversible effects of field inhomogeneities and the irreversible effects of nearest neighbor interactions with the reversible effects dominating. We now apply a $180^\circ$ pulse to reverse magnetization within the $x$-$y$ plane. At a time $\tau$ after the $180^\circ$ pulse all of the spins that were out of phase due to field inhomogeneities will be back in phase and an echo of the magnetization will appear in the detector where any loss of amplitude (relative to the amplitude of the FID following the initial $90^\circ$ pulse) is due only to the irreversible nearest neighbor interactions.

1. Establish resonance.
2. Make sure that $M_0$ is between 10 V and 10.5 V
3. Turn off the B pulse and trigger the scope on the A pulse.
4. Set the A Width to give a $90^\circ$ pulse.
5. Maximize the FID signal and find the magnet sweet spot.
6. Turn off the A pulse temporarily, turn on the B pulse and trigger the scope on the B pulse.
7. Set the B Width to give a $180^\circ$ pulse. Turn up the B width through its first maximum ($90^\circ$ pulse) to its first minimum which is a $180^\circ$ pulse.
8. Turn both A and B pulses on and set the scope to trigger on the B pulse.
9. For a short delay time $\tau$, view the echo and tune the B width to maximize the echo

NOTEBOOK and ANALYSIS:  > Measure the amplitude of the echo as a function of delay time. Take data for delay times ranging from the shortest time which will allow the echo to be resolved as separate from the FID, to the longest time for which the echo can be distinguished from the noise. The scope's cursor can be used to display the numerical value of the echo amplitude. $T_2$ can be extracted from a plot of the echo amplitude vs. delay time according to Eq. (28).

The viscosity of the mineral oil is strongly dependent on temperature. We also use a generic brand of commercial mineral oil and consequently there is no well established “accepted value” with which you can compare your measurements. Search the internet for values of time constants for other light and heavy oils in order to tell if your results are consistent with others. Beyond that however, you should treat the results of your measurements as you would a new measurement of a material for which the time constants are not known. In your analysis, you should strive to convince the reader that you have made an accurate measurement of the time constants for your mineral oil sample. You will find that it takes a fair amount of courage to present an experimental result, for which the “correct” value is not known, based solely on your interpretation of the data.

ANALYSIS:

Search the internet for time constant values for different oils (mineral or other types). Report a few values and discuss whether your results seem plausible compared to those findings.

The in-lab grade consists of three parts: a Day 1 question, a Day 2 question, and the lab notebook. Each rubric item below carries equal weight and the final grade for the in-lab portion (25% of the experiment grade) will be assigned based on the average over all rubric items.

As the TA looks over your notebook, they will be looking for general good practices and for specific things. Be prepared to point to (and explain) the appropriate places in your lab notebook where the following information is contained.

All rubric items carry the same weight. The final grade for the analysis (75% of the experiment grade) will be assigned based on the average (on a 4.0 scale) over all rubric items. Missing elements will receive a score of 0.