Table of Contents
Quantum Control (in development)
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In 1946 nuclear magnetic resonance (NMR) in condensed matter was discovered simultaneously by Edward Purcell at Harvard and Felix Bloch at Stanford using different techniques. Both groups observed the response of magnetic nuclei, placed in a uniform magnetic field, to a continuous wave radio frequency (RF) magnetic field as the field was tuned through resonance.
In 1950 Ervin Hahn, a young postdoctoral fellow at the University of Illinois, explored the response of magnetic nuclei in condensed matter to pulse bursts of these same RF magnetic fields. Hahn was interested in observing transient effects on the magnetic nuclei after the RF bursts. During these experiments, he observed a spin echo signal after a two-pulse sequence. This discovery, and his brilliant analysis of the experiments, gave birth to a new technique for studying magnetic resonance.
These discoveries and advances have opened up a new form of spectroscopy which has become one of the most important tools in physics, chemistry, geology, biology, and medicine. Magnetic resonance imaging scans (abbreviated MRI; the word “nuclear” was removed to relieve the fears of the scientifically uninformed public) have revolutionized radiology. This imaging technique is completely noninvasive, produces remarkable three-dimensional images, and gives physicians detailed information about the inner working of living systems.
Related Applications
- PNMR has industrial and agricultural applications in measuring the moisture content of substances
- PNMR can be used to determine properties of the structure of sedimentary rocks and the flow of liquids within them
Before you come to lab...
In order to prepare for the lab, you should read over the full theory section below, and complete the prelab exercises.
Motivation
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 water sample. These two time constants – and the techniques used to measure them – form the basis of medical MRI imaging.
At the same time, the system we will be investigating – the nuclear spin of a proton flipping between two well-defined states – represents a model system for a qubit. Through that lens, this experiment is also a study in how to prepare, manipulate, and readout such a qubit (which together represent the first steps in building a quantum computer).
1. Prelab exercises (5 points)
Read the theory below (which is rooted in the PNMR viewpoint). From this (and/or from other online sources), answer the following questions. They may seem quite challenging at first! We are looking for you to give them a good faith effort (and explain your reasoning), and it is OK if you do not get all the questions correct.
- How does a bar magnet (i.e. a macroscopic object with a fixed magnetic moment and no intrinsic angular momentum) behave when we turn on a static external magnetic field? In particular, consider the following:
- If the bar magnet is at rest and the magnetic moment is oriented in a certain (arbitrary) direction, what happens when the field is turned on?
- If we wait long enough for any short-term transient behavior to die out, does the orientation of the bar magnet's magnetic moment settle down so that it points in a single stationary direction or does it continue to move? Why?
- If we add energy (e.g. by grabbing it, rotating it, and trying to hold it in place), can we make it point in any arbitrary direction or are there only certain directions it can point?
- How does a proton (i.e. a microscopic spin 1/2 particle with a fixed magnetic moment and a non-zero intrinsic angular momentum) behave in a static external magnetic field? In particular, consider the following:
- If the proton is at rest and the magnetic moment is oriented in a certain (arbitrary) direction, what happens when the field is turned on?
- If we wait long enough for any short-term transient behavior to die out, does the orientation of the proton's magnetic moment settle down so that it points in a single stationary direction or does it continue to move? Why?
- If we add energy (e.g. through thermal excitation or through interactions with other particles), can we make it point in any arbitrary direction or are there only certain directions it can point?
- Suppose you have an ensemble of many protons in the same static external magnetic field. If the system is allowed to come to thermal equilibrium…
- What is the orientation of the bulk magnetization (the vector sum of the individual proton magnetic moments in the sample)? Does it remain static or does it change with time?
- What do you know about the orientation of any individual proton within the ensemble?
- Suppose now that you are able to temporarily take the ensemble of protons momentarily out of equilibrium by applying an external perturbation that changes the direction of the bulk magnetization. After the perturbation is turned off…
- What is the orientation of the bulk magnetization? Does it remain static or does it change with time?
- What do you know about the orientation of any individual proton within the ensemble?
In addition, please read this PDF which is a bit of a “teaser” of the apparatus put out by the manufacturer. There are no questions or assignments related to this reading, but it will help you see a little bit of what you are getting into with this experiment.
Theory
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 optical pumping.
We will first look at the general case of a spinning magnetic dipole moment in an external magnetic field and then connect that to the behavior of an individual proton.
Behavior of a single particle in an external magnetic field
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.
A spinning magnetic dipole in an external magnetic field
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
| $\boldsymbol{\tau} = \boldsymbol{\mu} \times \bf{B}$ | (1) |
that will act to align $\boldsymbol{\mu}$ with $\bf B$ and minimize the energy
| $E_{mag} = -\boldsymbol{\mu}\cdot\bf{B}.$ | (2) |
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| 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
| $\boldsymbol{\omega} = (\mu/L) {\bf B} = \gamma {\bf B}$, | (3) |
where we introduce the gyromagnetic ratio,
| $\gamma = \mu/L.$ | (4) |
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
| $\boldsymbol{\mu} = \mu_{xy}\cos{\left(\omega t\right)}\hat{\bf{x}} + \mu_{xy}\sin\left(\omega t\right)\hat{\bf{y}}+\mu_Z \hat{\bf{z}}$. | (5) |
A proton in an external magnetic field
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
| $E_{mag} = -\gamma {\bf I \cdot B} = -\gamma I_Z B_0 = -\gamma m_I \hbar B_0$, | (6) |
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 ($m_l = - 1/2$).
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| 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,
| $\Delta E = \hbar\omega_0 = \hbar\gamma B_0$, | (7) |
where we use the same relation between angular frequency, gyromagnetic ratio, and field as the classical bar magnet, Eq. (3):
| $\omega_0 = \gamma B_0$. | (8) |
For the proton, the resonant frequency, $f_0 = \omega_0 / 2\pi$, is
| $f_0 = 4.258 (\textrm{MHz/kG})B_0$, | (9) |
and its gyromagnetic ratio $\gamma$ is
| $\gamma_{proton} = 2.675 \times 10^4 \mathrm{rad/(s\cdot G)}$. | (10) |
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.
The PNMR technique is not sensitive enough to detect the behavior of individual protons; instead we work with the bulk properties of the sample which are the result of the behavior of the ensemble of protons. PNMR is most frequently used to make measurements of the relaxation times which characterize how long it takes for the nuclear spins belonging to an ensemble of protons to reach equilibrium. These time constants are referred to as $\mathrm{T}_1$ and $\mathrm{T}_2$.
- 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
- 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
Although we will focus specifically on protons (hydrogen nuclei), everything we discuss is generally applicable to other magnetic nuclei.
Behavior of an ensemble of protons in an external magnetic field: $T_1$ and $T_2$
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.
No external magnetic field
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.
Constant external magnetic field: Equilibrium state
If we now turn on an external magnetic field,
| ${\bf B} = B_0 \bf{\hat{z}}$, | (11) |
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,
| $\dfrac{N_1}{N_2} = e^{\frac{-\Delta E}{k_BT}} = e^{{-\hbar\omega_0/k_BT}}$, | (11) |
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,
| $\bf{M} =\displaystyle \sum_i \boldsymbol{\mu}_i$, | (12) |
we get a non-zero net magnetization along the $z$-axis,
| ${\bf{M}} = \left( N_1 - N_2\right) \mu = M_Z \hat{\bf{z}}$. | (13) |
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.
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| 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.
Measuring spin relaxation times
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:
- Create a known, non-equilibrium configuration of spin states.
- 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.
$T_1$: Spin-lattice relaxation time
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 ($T_1$):
| $\dfrac{dM_Z}{dt} = \dfrac{M_0 - M_Z}{T_1}$. | (14) |
Integrating this with the initial condition $M_Z (t=0) = -M_0$, we get
| $M_Z(t) = M_0(1-2e^{-t/T_1})$. | (15) |
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| 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. |
$T_2$: Spin-spin relaxation 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
| $\dfrac{d\vec{M}_{x*}}{dt} = -\dfrac{\vec{M}_{x*}}{T_2}$ | (16a) |
and
| $\dfrac{d\vec{M}_{y*}}{dt} = -\dfrac{\vec{M}_{y*}}{T_2}$, | (16b) |
the solutions of which are
| $M_{x*}(t) = M_0 e^{-t/T_2}$ | (17a) |
and
| $M_{y*}(t) = M_0 e^{-t/T_2}$. | (17b) |
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| 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. |
Apparatus and Procedure
While this lab is in development, we will be working out of the manufacturer's instruction manual. You do not need to read all of this before coming to lab, but feel free to start looking through it.
Final data analysis
You have one week after the last day in lab to perform a full and complete analysis of the data you collect and to submit the following assignments. You can score up to a maximum of 75 points total on these assignments.
Data collection and motivation (25 points)
As a first step, describe your data collection and the motivation behind your measurements. (I.e., Here's what I did.) We are not looking for a formal theory or procedures section, but instead want a description of the physics behind each measurement (both the quantity or phenomenon that that you are trying to investigate/manipulate and the physics involved in using your apparatus or technique). Put broadly, for this section you will be evaluated on…
- How did you collect data?
- Why did you collect the data you did?
- What did you expect to find (or hope to explore) when you collected that data?
Do the following:
- Describe how you prepared the apparatus to make your first measurements and how you optimized the signals you detected.
- Include photos, scope traces, physical motivation, and measurements as needed to support this preparation discussion.
- Describe what it means to apply a $\pi/2$ pulse, a $\pi$-pulse, etc. (both in terms of what you do with the apparatus and what the effect(s) on the protons in the material are).
- Include photos, scope traces, physical motivation, and measurements as needed to support this preparation discussion.
- Describe each separate experiment that you ran. Motivate the phenomenon you are exploring or the quantity you are trying to measure, and fully explain your experimental approach to exploring or measuring it.
Data presentation (25 points)
Next, present the data that you collected for each of the experiments you ran (i.e. Here's what I saw). Put broadly, for this section you will be evaluated on…
- How did you present your findings (including plots, tables, scope traces, and figures)?
- How did you contextualize the results within the text?
- Is your presentation clear and easy to follow for the reader?
Do the following:
- Provide raw and computed data (as appropriate) for each experiment you ran in an appropriate format.
- Explain how you estimated statistical uncertainties.
- Explain how you looked for systematic uncertainties (and, if appropriate, how you explored, quantified, or eliminated those uncertainties and/or biases).
Discussion (25 points)
Finally, you will want to discuss your results and provide appropriate context for the reader (i.e. Here's what it means). Put broadly, for this section you will be evaluated on…
- How do your results compare to either literature values (provide citations) or to your expectations (provide reasoning)?
- How do uncertainties (both statistical and systematic) affect your confidence in the results?
- Specifically, How would you improve your data collection (either to fix biases or to reduce uncertainties) in order to improve your confidence in the results?
- Are your discussion and any conclusions you draw supported by the results, and is your presentation clear and easy to follow for the reader?
Note: While this experiment is still in development, there may not yet be a neat “conclusion” (since there wasn't a clear goal from the start). That said, you still need to provide a coherent discussion of what to take from the results.
Do the following:
- Highlight the most important take-away quantities and make sure there is a clear, final message.
- Discuss the agreement between the data and the literature (for individual points and/or overall, as appropriate).
- Justify your results and discuss the context surrounding your measurements. This may include asking yourself the following non-exhaustive list of questions:
- What do your results mean?
- E.g., for someone who is hearing about this experiment for the first time, why are your results important and why is this experiment the right way to make these measurements?
- Why should the reader believe your results and your estimates of uncertainties on those values?
- E.g., what about your technique and data collection or analysis strategy gives the reader confidence in your work?
- If you disagree with expectations, do you disagree in some systematic way or is it random?
- E.g., consider your results in the larger physics context, and address anything which isn't consistent with what you (and other experts) know. Separate statistical uncertainty from systematic bias.
- Is your experiment complete, or are there holes in the picture?
- E.g., are there results that you cannot explain or disagreements that you cannot give plausible explanations for? Do you need more (or different data) to investigate the problem fully, or are there measurements that need to be retaken or rethought?
- Consider shortcomings in your work or things that you would improve if you were to continue this experiment (e.g. changes apparatus or technique, alternate analysis methods, or different models). Point towards future directions of study.
- Be specific! Do not just say “more data” or “better equipment”. Justify the suggestions you make.
References (Quantum Control)
References (PNMR)
[1] Pulsed NMR Apparatus Manual, TeachSpin Inc.
[2] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York, 1996.
[5] E. L. Hahn, "Spin Echoes", Physical Review 80(4), 580 (1950).
Experimental Technique
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 water 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 reorient 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 $\mathrm{T}_1$ 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 $\mathrm{T}_2$.
3. Measurements of the bulk magnetization are performed as the spins relax back towards equilibrium.
We will now discuss how our apparatus accomplishes this.
Experimental procedure
Observing the Free Induction Decay (FID) Signal
Make sure the controls of the pulse programmer are set as follows:
- A-width: At about the 9 O'Clock position. (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
- Variable knob: 100% (pulses should be produced 100 ms * 100% = 100 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 Glycerin sample. Connect the Detector Out from the 15 MHz receiver to channel 1 of the scope. This 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.
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| 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. | ||
Connect the Mixer Out port of the Teachspin unit to channel 2 of the oscilloscope. Adjust either the magnetic field strength or the reference RF frequency until the beat frequency drops (close) to zero (as seen by observing the Mixer signal). At this point, we know that the protons are precessing at a frequency equal to the reference frequency.
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.
Optimizing the FID
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.
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.
Understanding the FID
Now that you have obtained a good FID signal, we want you to perform the following exercises to make sure that you have sufficient understanding of what the FID is, and what it represents in terms of what is going on with the protons in the sample, that you will be able to properly setup and execute the multi-pulse sequences required for the $T_{1}$ and $T_{2}$ measurements.
Duration of a 90° pulse
What we are calling the FID is the signal induced in the detector coil immediately after a 90° pulse has been applied to the ensemble of protons in the equilibrium state. When you set the “A-Width” on the pulse programmer you are specifying how long the RF Oscillator is on and generating a ~15MHz rotating B-field at the sample. If you understand what a 90° pulse is and the role played by the RF Oscillator is, you should be able to calculate how long the RF-Oscillator needs to be on in order to induce a 90° rotation of the protons in the sample.
When it is on, the RF-Oscillator produces a 12 Gauss rotating field at the sample. You know that the precession frequency of the proton is determined by $\boldsymbol{\omega} = (\mu/L) {\bf B} = \gamma {\bf B}$.
Calculate how long the RF-Oscillator needs to be on in order to cause a 90° rotation of the protons in the sample. Then use the scope to directly measure the duration of the A-Width pulse to confirm your prediction. For now, Use the literature value for the gyromagnetic ratio from the bottom of this page.
Confirm your understanding by repeating this exercise for a 180° rotation of the protons in the sample.
Without doing anymore calculations or measurements on the scope, it should be trivial to say how long the A-Width pulse needs to be set for a 270° and a 360° pulse.
Repetition time and $T_{1}$ relaxation
The repetition time setting on the pulse programmer simply determines the time interval between successive A pulses (or A and B pairs if you have a B pulse turned on). When setting up to measure relaxation times it is possible to set the repetition time too short, so that the system does not have enough time to completely relax back to equilibrium after an A (or AB) pulse, before the next A (or AB) pulse comes along. If you understand the concepts of a 90° pulse and $T{1}$, you should be able to predict what will happen to the FID if the repetition time is set too short.
Use the pulse programmer and Scope to confirm your prediction for how the FID should behave when the repetition time is reduced. Note this is a qualitative exercise, you do not need to calculate anything.
Once you understand how the FID behaves as you reduce the repetition time, you should be able to use this effect to measure (within a factor of 2) the $T{1}$ relaxation time for your 100% Glycerine sample. Do this.
You should not proceed to the next section until you have both completed and understand the above exercises. If necessary find one of the lab staff to assist you in this.
Measurements
Measuring the gyromagnetic ratio for protons and fluorine nuclei
While at resonance (zero-beat condition) remove the sample tube containing the Glycerin 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.
From your measured values of resonant frequency and magnetic field, calculate γ for protons. Because of the imprecise manner in which we are measuring the magnetic field at resonance, by using the hall effect gaussmeter, this is not going to be a highly precise measurement.
Gyromagnetic ratio of protons in teflon
Repeat the measurement for Fluorine atoms in the teflon sample.
Fluorine atoms are what give teflon its non-stick properties. Because of its different gyromagnetic ratio you will have to increase the magnetic field for the teflon sample in order to see the FID.
Also solid materials tend to produce smaller amplitude signals with much faster decay times. So the FID can be difficult to find, you can make your life easier by estimating how much the magnetic field needs to increase relative to protons.
 (A) |  (B) |
| Figure 20. (A) DC Gaussmeter (B) calibration magnet set | |
Measuring $T_{1}$ and $T{2}$ for Glycerine
The next two section describe the Inversion Recovery and Spin Echo methods of measuring $T_{1}$ and $T{2}$ respectively. You have three sample vials each with a different Glycerine to Water ratio. The viscosity of Glycerine changes significantly with the addition of small amounts of water. Higher viscosity implies more interaction between particles in the solution, which should result in shorter relaxation times. Measure both $T_{1}$ and $T{2}$ for each of the samples. Pure Glycerine, which has the highest viscosity, is the easiest of the three samples to work with, so we suggest you begin with that sample.
Measuring T1 with the inversion recovery method ($180^\circ \rightarrow \tau \rightarrow 90^\circ$ pulse sequence)
Measure T1 for the Glycerin samples only. You do not need to measure it for Teflon.
Also, record your (constant) magnetic field here, T1 is sensitive to its specific value.
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, $\tau$:
| $M_Z(t) = M_0(1-2e^{-t/T_1})$. | (15) |
Procedure:
- 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.
- Set the A pulse to $180^\circ$, the first minimum.
- Turn off the A pulse, turn on the B pulse and trigger the scope on the B pulse*.
- Set the B width to give a $90^\circ$ pulse by maximizing the first FID signal.
- Turn both A and B pulses on and set the scope to trigger on the B pulse.
- Optimize the output signal by checking the magnet sweet spot and small tuning of the A and B pulse widths.
- Adjust the delay time settings. What effect does this have on your output signal?
If you set the delay time (how long it takes for the second pulse to be sent) to be longer than the repetition time (how long before the entire pulse sequence repeats), the pulse programmer may behave strangely and may not properly create the B pulse. If you have triggering issues for the B pulse, try adjusting these variables.
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 $e^{-t/T_1} = 1/2$?
Measuring T2 using the spin echo method
Measure T2 for the Glycerin sample only. You do not need to measure it for Teflon.
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.
For the rest of this discussion we assume that the effects of the electromagnet inhomogeneities on proton precession differences are constant over the course of any single measurement. In other words protons in any particular region of the electromagnet field remain in that region and do not migrate into a region of substantially different field strength. This assumption does NOT hold however for precession frequency differences caused by nearest neighbor spin-spin interactions which fluctuate randomly on time scales short compared to the time it takes to make a single 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
| $M_{x,y}(2\tau) = M_0 e^{-2\tau /T_2}$. | (28) |
The spin echo method due to Hahn uses a pulse sequence $90^\circ \rightarrow \tau \rightarrow 180^\circ \rightarrow \tau \rightarrow$ “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.
- Establish resonance.
- Make sure that $M_0$ is between 10 V and 10.5 V
- Turn off the B pulse and trigger the scope on the A pulse.
- Set the A Width to give a $90^\circ$ pulse.
- Maximize the FID signal and find the magnet sweet spot.
- Turn off the A pulse temporarily, turn on the B pulse and trigger the scope on the B pulse.
- 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.
- Turn both A and B pulses on and set the scope to trigger on the B pulse.
- For a short delay time $\tau$, view the echo and tune the B width to maximize the echo
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 Delay Time selector. Note the decimal between the first two digits and the last number indicating an exponent. In this instance, the delay time is $1.23\times10^1 \mathrm{ms} = 12.3 \mathrm{ms}$ |
When you leave the lab
Since there will be other groups working on the apparatus it is your responsibility to ensure that everything in the lab is in order with the next group arrives. The room should be tidy and everything should be either put away or reset to it's default. If the lab room was in disarray when you arrived you are still responsible for leaving it in the appropriate state for the next group.. Here are some general tips on things to check before you leave.
- The electromagnet power supply should be dialed down to zero and shut off.
- The water to the electromagnet should be turned off.
- The sample vials should be returned to their holder.
- Any cabling changes you may have made to the TeachSpin electronics should be reset to the default condition.
- The TeachSpin electronics should be returned to the default settings and turned off.
- The scope should be turned off and disconnected from the computer.
- The room lights should be turned off.
- Close any applications on the computer. Make sure to log out of any accounts which you may have logged into. The computer can be left on, it will go into sleep mode.
- Sign out of the logbook.
Analysis and Final Report
The analysis is not a lab report, rather it is all of the data reduction, number crunching, calculations, curve fitting, error propagation etc. which is necessary for you to establish your final conclusions. Think of it as being more like an extended homework set where you have to show how you got your final results.
Three days after your analysis submission your group will have a meeting with the TA to go over your analysis and make sure you are prepared to write your final report.
Your graded analysis will be returned along with your graded final report.
Analysis
Shortly after your analysis is due your group will meet with the TA to discuss the overall analysis and make clear what needs to go into your final report. Note that this meeting is not for the purpose of discussing your grade on the analysis, you will receive the grade on the analysis along with the graded final report. Instead this is an opportunity for the TA to have reviewed your analysis to identify where you may have short comings or misconceptions in your understanding of the experiment with the goal of improving what goes into your final report. It is also an opportunity for you to make sure that you understand what your TA is looking for in your report.
Your analysis, like your reports, should be submitted as a single PDF. It is not expected that you will write narrative descriptions as you will in your final report. For the analysis it is acceptable to organize it into sections with one or two brief sentences of description. Things should be put in a sensible order so that the TA can follow what you are doing. For example plots, fits and calculations related to your energy calibration should be grouped together into a section, and that section should be placed before you apply the calibration to your data. For cases such as fitting and extracting peak locations for all of your scattering data it is sufficient to show one representative plot of a fit to the data along with a table containing all of the values. Scans or photographs of calculations done on paper or in your lab notebook are acceptable but absolutely MUST be clear and readable.
Final Report
Your final report will be evaluated based on the following rubric. The rubric is not a format for your analysis, you are not expected to have a specific section on Data Handling or Presentation of Data. Elements of the different rubric categories will appear at different points through out your analysis writeup. For example you will be presenting data in your discussion of the calibration, your discussion of determining peak locations, and likely in your final results. Your writeup of your analysis should be structured in a way that is clear and readable, there should be a logic to the flow of it.
Each item below is graded on a 0-4 point scale:
- 4 – Good (A): completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
- 3 – Adequate (B): misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
- 2 – Needs improvement (C): omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
- 1 – Inadequate (D): omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
- 0 – Missing (F): omits all elements or makes no meaningful attempt.
All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.
| Item | Good (4) |
| Flow | The report is well organized and clearly written. The logical flow of how information is presented makes it easy for the reader to understand what is being communicated. Extraneous information unrelated to the conclusions is minimized. |
| Presentation of Data | Presents plots of data as needed and uses them to support the narrative of the report. Properly labels plots, and makes presentation clean and clear. Uses error bars where appropriate. Includes captions that provide appropriate context. Presents all numerical values with appropriate units and significant figures. Clearly formats numbers, equations, tables, etc. |
| Data Handling | Describes how the raw data was processed including with uncertainties. Details fit functions and provides sample fits (if appropriate). Details other calculations/considerations and provides sample calculations or reasoning (if appropriate). |
| Discussion of Uncertainties | Identifies relevant sources of uncertainty in measured quantities, and quantifies values when possible. Describes how uncertainties were assessed and incorporated into the analysis. Identifies potential sources of systematic bias and describes how they are accounted for in the analysis or eliminated. |
| Presentation of Results | Final results are presented clearly. Data tables and plots are used where appropriate and are properly labeled and annotated. Measured and calculated quantities include units and uncertainties where appropriate. |
| Conclusions | Makes clear final conclusions that are fully supported by the experimental results and discusses the overall take-aways of the experiment appropriately. Properly accounts for or contextualizes measurement uncertainties and potential sources of systematic bias. |
References
[1] Pulsed NMR Apparatus Manual, Teach Spin Inc.
[2] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York, 1996.
[5] E. L. Hahn, "Spin Echoes", Physical Review 80(4), 580 (1950).