Pulsed Nuclear Magnetic Resonance: Spin Echoes

MIT Department of Physics(Dated: September 19, 2017)

This experiment explores nuclear magnetic resonance (NMR) both as a physical phenomenonconcerning atomic nuclei and as a ubiquitous laboratory technique for exploring the structure ofbulk substances. Using radio frequency bursts tuned to resonance, pulsed NMR perturbs a thermalspin ensemble, which behaves on average like a magnetic dipole. One immediate consequence is theability to measure the magnetic moments of certain nuclei such as hydrogen (i.e., the proton) andflourine; the former is of particular interest to nuclear physics.

In addition, the use of techniques like spin echoes lead to a myriad of pulse sequences which allowthe determination of spin-lattice and spin-spin relaxation times of substances. Among the samplesavailable in this lab are glycerin and paramagnetic ion solutions, whose viscocity and concentrationstrongly affect their relaxation times. Investigation of these dependences illustrate the use of pulsedNMR as a method for identifying and characterizing substances.

PREPARATORY QUESTIONS

Please visit the Pulsed NMR chapter on the 8.13r web-site at lms.mitx.mit.edu to review the background ma-terial for this experiment. Answer all questions found inthe chapter. Work out the solutions in your laboratorynotebook; submit your answers on the web site.

PROGRESS CHECK

By the end of your 2nd session in lab you should havea determination of the nuclear magnetic moment of fluo-rine. You should also have a preliminary value of T2 for100% glycerine.

I. BACKGROUND

The NMR method for measuring nuclear magnetic mo-ments was conceived independently in the late 1940sby Felix Bloch and Edward Purcell, who were jointlyawarded the Nobel Prize in 1952 for their work [1–4]Both investigators, applying somewhat different tech-niques, developed methods for determining the magneticmoments of nuclei in solid and liquid samples by mea-suring the frequencies of oscillating electromagnetic fieldsthat resonantly induced transitions among their magneticsubstates, resulting in the transfer of energy between thesample of the measuring device. Although the amountsof energy transferred are extremely small, the fact thatthe energy transfer is a resonance phenomenon enabledit to be measured.

Bloch and Purcell both irradiated their samples with acontinuous wave (CW) of constant frequency while simul-taneously sweeping the magnetic field through the reso-nance condition. CW methods are rarely used in modernNMR experiments. Rather, radiofrequency (RF) energyis usually applied in the form of short bursts of radiation(hence, the term ”pulsed NMR”), and the effects of theinduced energy level transitions are observed in the time

between bursts. More important, as we shall see, thetechniques of pulsed NMR make it much easier to sortout the various relaxation effects in NMR experiments.

Nevertheless, this experiment demonstrates the essen-tial process common to all NMR techniques: the detec-tion and interpretation of the effects of a known perturba-tion on a system of magnetic dipoles embedded in a solidor liquid. As we shall see, this analysis of the system’sresponse to what is essentially a macroscopic perturba-tion yields interesting information about the microscopicstructure of the material.

II. THEORY

II.1. Free Induction of a Classical MagneticMoment

In classical electromagnetism, a charged body withnonzero angular momentum L possesses a quantity calleda magnetic moment µ, defined by

µ = γL, (1)

where γ is the body’s gyromagnetic ratio, a constantdepending on its mass and charge distribution. For aclassical, spherical body of mass m and charge q dis-tributed uniformly, the gyromagnetic ratio is given by

γcl =q

2m. (2)

The magnetic moment is an interesting quantity be-cause when the body is placed in a static magnetic fieldB0, it experiences a torque

dL

dt= µ×B0. (3)

This equation of motion implies that if there is somenonzero angle α between L and B0, its axis of rotationL precesses about B0 at the rate

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ωL = −γclB0 (4)

independently of the value of α, much like the behaviorof a gyroscope in a uniform gravitational field. The mi-nus sign indicates that for a body with positive γ (e.g.,positively charged sphere), the precession is clockwise.This phenomenon is called Larmor precession, and wecall ωL =| ωL | the Larmor frequency.

Suppose B0 = B0z; in this case, the x-y plane iscalled the transverse plane. Next, suppose we place asolenoid around the magnetic moment, with the axis ofthe solenoid in the transverse plane (e.g., aligned withx). If α is nonzero, there is a nonzero transverse compo-nent of µ, which generates an oscillating magnetic field atthe Larmor frequency. By Faraday’s law, this transversecomponent induces an emf

V (t) = V0(α)cos(ωLt+ φ0). (5)

The phase φ0 is simply the initial transverse angle be-tween the µ and the solenoid axis, while V0 is an overallfactor that incorporates constants like the overall magni-tudes of µ, signal amplification, and solenoid dimensions.It is worth noting, however, that V0 is a function of α:when α is zero, there is no signal, and as α is moved to-wards π/2, the signal reaches a maximum when µ lies inthe transverse plane, which then decreases back to zeroat α = π when µ is antiparallel to B0, and so on.

But regardless of its magnitude, the induced emf al-ways oscillates at the characteristic Larmor frequency.We call this detected voltage oscillation the free induc-tion NMR signal. It is this signal with which NMR is pri-marily concernedwe manipulate the magnetic moment ofa sample and monitor the behavior of the free inductionsignal to understand its bulk material properties.

II.2. Nuclear Magnetism

In quantum mechanics, it is a fact that particles (i.e.,electrons, protons, and composite nuclei) possess an in-trinsic quantity of angular momentum known as spin,which cannot quite be understood as any form of classicalrotation. The particle’s spin angular momentum alongany given direction is quantized, andfor a spin − 1

2 parti-cletakes on the values of +~/2 and −~/2, correspondingto two states we usually refer to as ”spin-up” and ”spin-down”. The general state (wavefunction) of any suchtwo-state system is a complex superposition of these twoeigenstates. We sketch a rough picture below of howmacroscopic nuclear magnetism comes out of this micro-scopic framework; for more accurate details, see [5, 6].

There is no reason we should expect such a systemto behave anything like a classical particle with angularmomentum. Yet, as shown in Appendix A, the wave-function for any two-state system can be visualized as a

vector in the Bloch sphere, using the relative phases inthe superposition as direction angles. And it turns outthat in this picture, when a charged quantum spin suchas an atomic nucleus is placed in a static magnetic field,the wavefunction does indeed exhibit an analogue of Lar-mor precession in the Bloch sphere. Of course, there arekey differences. For one thing, the precession frequencyis not the same, owing to quantum effects. The differenceis captured by changing the gyromagnetic ratio:

γ = gγcl. (6)

This corrective g-factor is analogous to the Lande g-factor in atomic spectroscopy and varies from nuclei tonuclei. We retain the definition of the magnetic mo-ment, so that in an eigenstate of angular momentum,the magnitude of the magnetic moment in that directionis µ = γ~/2.

More significantly, however, precession of the wave-function in the Bloch sphere is not quite the same asa precession in real space. For example, with a classi-cal magnetic moment µ aligned at some nonzero angle αwith z, it is possible to measure precisely both µx = µ · xand µz = µ · z. But Heisenberg’s uncertainty principleforbids simultaneous measurements of both quantum op-erators and µx, µz even though the wavefunction vectorin the Bloch sphere has well-defined projections onto xand z.

Nevertheless, the expectation of the quantum magneticmoment does behave exactly like the classical magneticmoment, in the limit of a large number of repeated mea-surements (a case of Ehrenfest’s theorem). It is preciselythis correspondence that NMR relies on. But rather thanmaking repeated measurement of a single quantum spin,we make an ensemble measurement of a large number ofspins at once. If their wavefunctions are approximatelythe same (i.e., the spins are ”coherent”), then the en-semble should exhibit a macroscopic, classical magneticmoment.

More precisely, what we measure in NMR is M = N〈 µ 〉 , or the expectation of the quantum magnetic mo-ment averaged over the bulk ensemble, multiplied by thenumber of spins which are coherent. Since the volume ofour sample is fixed, this is proportional to the magnetiza-tion, and we will simply refer to M as the ”magnetizationvector” of the sample. Our conclusion is that M exhibitsthe exact same dynamics as the classical magnetic mo-ment µ , generating an NMR free induction signal at theLarmor frequency ωL = γB0. It is worth noting that thismacroscopic ensemble does indeed capture the quantumnature of the spins, by way of the non-classical gyromag-netic ratio γ.

II.3. Pulsed NMR

As suggested in Problem 3, the equilibrium state fora system of spins in a static magnetic field B0 = B0z

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produces a small magnetization M0 = M0z in the samedirection as the magnetic field. However, we also knowthat such a configuration in which α = 0 still does notresult in a free induction signal. We also need a wayof perturbing the spins out of equilibrium, to generatea transverse component of the magnetization. We use amethod called pulsed NMR in order to achieve this.

In pulsed NMR, the same solenoid that is used to pickup a transverse magnetization can also be used to gen-erate an RF field around the sample. We focus on thegeneration of a magnetic field

B1(t) = B1 cosωtx (7)

It is standard to take B1 << B0, so that the field B1

can be treated as a perturbation. We then proceed toexamine the behavior of M in response to this perturbingfield.

We first note that we can write B1 as a superpositionof two counter-rotating magnetic fields:

Br(t) =B1

2(cosωtx+ sinωty) (8)

Bl(t) =B1

2(cosωtx+ sinωty) (9)

Thus, B1 = Br + Bl. For nuclei with positive gyro-magnetic ratio, Bl rotates in the same direction as themagnetization (clockwise), while Br rotates in the oppo-site direction (counter-clockwise).

We now consider the situation from the point of view ofan observer in a reference frame rotating in the directionof precession (that is, clockwise) with angular velocity ω.The unit vectors in this rotating frame are

x′ = cosωtx− sinωty (10)

y′ = sinωtx− cosωty (11)

z′ = z (12)

A moment’s thought will confirm that in the x y coor-dinate system, the x′ y′ system is indeed rotating clock-wise with angular velocity ω.

In this rotating frame, the field Bl appears to bestationary, while Br appears to be rotating counter-clockwise at a rate 2ω. This can be shown directly bysolving for x and y in terms of x′ and y′ in the equa-tions above and substituting. The result is that the tworotating components become

Br =B1

2(cos 2ωtx′ + sin 2ωty′) (13)

Bl =1

2Blx′ (14)

On the other hand, the static magnetic field does notappear any different, and B0 = B0x′.

The total magnetic field in the rotating frame is, ofcourse, B = B0+B1. But this magnetic field shouldevenin the rotating frameinduce a Larmor precession. Theprecession angular velocity of the magnetization vectorin this frame is

Ω = −γB + ωz′ (15)

where the extra term comes from the kinematic mo-tion of the rotating frame (as if there were a fictitiousmagnetic field opposing B0 ). Written more explicitly interms of components, we have

Ω · x′ = −γB1

2− γB1

2cos 2ωt (16)

Ω · y′ = −γB1

2sin 2ωt (17)

Ω · z′ = −γB0 + ω (18)

Now the crucial point: when the frequency of the per-turbing field satisfies ω = γB0 = ωL (on resonance withthe natural Larmor frequency), the rapid precession duetoB0 vanishes in the rotating frame, and all that remainsis a constant, slow precession about x′ at the rate γB1/2,with only the addition of tiny time-dependent flutters(the sines and cosines), which average out to zero.

If M is initially parallel to B0, then application of theperturbing pulse B1 for a time

t90 =π

γB1(19)

evidently rotates M by 90 about x′, placing M in thetransverse plane perpendicular to B0. If the pulse is nowturned off, M is left in the transverse plane, and fromthe point of view of an observer in the laboratory frame,it will be precessing at the Larmor frequency γB0 aboutz. By a similar argument, application of the perturbingpulse for a time t180 = 2t90 rotates M by 180, invertingthe spin population. In practice, the value of 1 is notwell-known, so t90 is usually found by trial and error,usually by looking for the pulse width which yields thegreatest transverse magnetization.

II.4. Relaxation

Owing to the microscopic nature of nuclear magnetism,the free induction signal does not persist for very long.

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FIG. 1. An idealized scope trace of a free induction decaysignal, showing also the decay envelope. The thick black lineindicates the 90 perturbing pulse that puts the magnetiza-tion into the transverse plane. The decay constant T ∗

2 consistsof both the T2 effect discussed below as well as the effect offield inhomogeneities (discussed next section). Due to thelatter effect, T ∗

2 >> T1 in the real NMR setup.

Once perturbed, the spins proceed to return towardsequilibrium, in a process called relaxation. Relaxationis one of the keys to the utility of NMR: different sub-stances return to equilibrium at different rates and indifferent ways; analysis of the relaxation times of a sam-ple gives significant insight into its chemical compositionand structure.

Relaxation mechanisms (and other effects, as discussedin the following section) result in an exponential decayin the free induction NMR signal, which manifests itselfas the ubiquitous free induction decay (FID) signal, asketch of which is shown below.

There are two relaxation mechanisms which are ofphysical interest in this lab. The first is the eventualrecovery of longitudinal magnetization (that is, magneti-zation along B0 ), due to rethermalization of the system.The second, which occurs even in the absence of the first,is the loss of transverse magnetization due to decoherenceof the spins. Both of these mechanisms contribute to thedecay observed in the FID, and the rates at which theyoccur depend on the substance in question.

The first mechanism is typically called spin-lattice re-laxation and the time constant governing its rate is de-noted T1. Its name derives from the fact that rethermal-ization is caused by the redistribution of energy from thespins to their surrounding environment (the ”lattice”).This has the effect of dissipating the energy of the pulseuntil the entire sample has returned to its original ther-mal state. If we use a 90 pulse to send the magnetizationinto the transverse plane at t = 0, then the spin-latticerelaxation process can be described by saying Mz recov-ers according to

Mz(t) = M0(1− e−t/T1), (20)

where M0 is the magnitude of the longitudinal mag-

netization at thermal equilibrium. Of course, as Mz re-covers, the transverse magnetization correspondingly de-creases.

The second mechanism is typically called spin-spin re-laxation and the time constant governing its rate is de-noted T2. Its name derives from the idea that, as thespins are precessing, they feel small fluctuations in themagnetic field from magnetic dipole interactions of neigh-boring spins (perhaps other nuclei on the same molecule),which leads to randomization of the spin’s precessionalmotion. This causes the spinswhich were initially precess-ing in phase and constructively contributing to the trans-verse magnetizationto decohere and begin destructivelyinterfering, diminishing the observed transverse magne-tization. This spin-spin relaxation process, like the spin-lattice relaxation, is an irreversible process, and it canbe described by saying that the transverse magnetizationMxy decays according to

Mxy(t) = M0e−t/T2 (21)

where M0 is the initial transverse magnetization at t= 0 right after a 90 pulse.

Most of the measurement techniques used in this labcenter on the goal of obtaining these relaxation times forvarious substances. In particular, we are interested inhow these relaxation times change as we vary the prop-erties of the substance such as concentration or viscos-ity. Typically, experiments to determine T1 perturb thesystem, let it relax, and then attempt to measure the re-covered longitudinal magnetization. On the other hand,experiments to find T2 rely on making observations ofhow the FID signal decays.

II.5. Spin Echoes

Although we are primarily interested in relaxation,there are other effects that contribute to the decay inthe FID. The most important of these is inhomogene-ity in the magnetic field. A global inhomogeneity in thestatic magnetic field B0 can cause parts of the sampleto precess at different rates, leading to phase differencesand a loss of the ensemble-averaged transverse magneti-zation much more quickly than would be expected fromjust spin-spin interactions alone.

In fact, for simple NMR setups such as the one used inthis lab where the static magnetic field is maintained bypermanent magnets, such field inhomogeneities dominaterelaxation. The observed decay constant of the FID istypically denoted T ∗

2 , and it consists of two components:

1/T ∗2 = 1/T2 + γ∆H0, (22)

where T2 is the spin-spin relaxation time and ∆H0 is ameasure of the inhomogeneity of the magnetic field overthe sample volume.

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FIG. 2. An idealized scope trace for a spin-echo sequence,in a setup where T2 >> T ∗

2 . The thick black lines indicatethe perturbing pulses used to implement the pulse sequence.Notice the spin echo is produced at time 2τ .

Fortunately, however, the effect of field inhomo-geneities is to some extent a reversible process; it is moreof a dispersion effect than decoherence. Even after theFID has decayed away, it is possible to recover the trans-verse magnetization, up to whatever amount has beenirreversibly lost in relaxation. This recovery was discov-ered by Erwin Hahn in 1950 and is known as a spin echo.

The spin echo pulse sequence can be described as90τ180, or a 90 pulse, after which the FID is allowedto decay away for time τ , at which point a 180 pulseis applied. A spin echo forms at a time τ after this lastpulse, as shown in the figure below.

To see how the spin echo is produced, consider a typi-cal sample which has small regions of uniform magneticfield, but such that the field differs from one region tothe next. Following a 90 pulse, spins in a region of rela-tively high magnetic field precess faster, while those in aregion of relatively low magnetic field process slower. Bya time τ later, the phases of the magnetization across dif-ferent regions disagree sufficiently to degrade the overallmagnetization.

But the spins within each individual region are stillcoherent and precessing in the transverse plane. Theapplication of a 180 pulse has the effect of reflectingthese transverse spins about the direction of the appliedpulse. The spins continue to precess, but their relativemotion is now precisely reversed. Thus, those regionswhich were precessing faster and accumulated more phasedifference now undo their phase accumulation at a fasterrate. The result is that a time τ after the 180, all theregions are back in phase and the total magnetizationreaches a maximum, producing a spin echo.

The use of a spin echo allows us to obtain dispersion-free access to the transverse magnetization. Looking atthe height of the spin echo generated effectively tells uswhat the amplitude of the FID would have been at time2π if field inhomogeneity were not present. If T ∗

2 <<T2, which is usually the case in setups like ours, this

FIG. 3.

is information that would have been difficult to obtainwithout the spin echo technique.

Of course, there are limitations to the technique. It ispossible for spins in one region of uniform magnetic fieldto diffuse randomly to another. If this diffusion happenswithin the duration 2π required to execute the spin echopulse sequence, then the precise dephasing process wedescribed would no longer hold, and the spin echo am-plitude would be reduced beyond just relaxation. Carrand Purcell in 1954 showed [7] that when we add in theeffects of diffusion, the echo amplitude produced goes as

E(2π) = E0exp(−2τ

T2− 2

3γ2G2Dτ3), (23)

where E0 is the echo amplitude in the absence of bothspin-spin and diffusion effects (i.e., the initial FID ampli-tude). Here, G is the gradient of the inhomogeneous fieldand D is the diffusion constant. Thus, the effect worsensthe longer we wait to produce a spin echo.

III. APPARATUS

III.1. Apparatus Overview

The experimental apparatus, shown schematically inthe following figure, consists of a gated RF pulse gen-erator with variable pulse widths and spacings, a probecircuit that delivers RF power to the sample and picksup the signal from the sample, a preamp that amplifiesthe signal, and a phase detector which outputs an au-dio signal whose frequency corresponds to the differencebetween the Larmor frequency and the frequency of thesignal generator. Details of how to design and build NMRprobes can be found in R. Ernst and W. Anderson, Rev.Sci. Instrum. 37, 93 (1966).

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FIG. 4.

III.2. Permanent Magnet

This experiment uses a permanent magnet whose fieldis about 1770 gauss (0.177 Tesla). Care should be takento avoid bringing any magnetizable material (such as ironor steel) near the magnet as this may be pulled in anddamage the magnet.

When performing the experiment, you should try tofind a region where the magnetic field is most uniform toinsert your sample and label the position of the probe forreproducibility between experiment runs.

III.3. RF Signal Chain

Although it is the policy in Junior Lab to discour-age the use of pre-wired experiments, there are tworeasons why the present set-up should not be (lightly)changed. Several of the components, particularly thedouble-balanced mixers (DBM) and the low-level TRON-TECH pre-amplifier, are easily damaged if the RF powerlevel they are exposed to exceeds their specified limit.Furthermore, the lengths of some of the cables have beenspecifically selected to fix the relative phase relationshipof different signals.

The RF pulse generating system is made up of a 15MHz frequency synthesizer (Agilent 33120A), a digitalpulse programmer based on a STAMP micro-controller,a double-balanced mixer used as an RF switch (Mini-Circuits ZAS-3), a variable attenuator, and an RF poweramplifier capable of 2 watts output.

The frequency synthesizer feeds a +10 dBm RF sinewave to the power splitter. The power splitter keeps allimpedances appropriately matched while feeding one halfof the RF power to a double-balanced mixer (DBM) usedas a gate for the RF. The other half is used as a refer-ence signal in the phase detector. The gate is openedand closed by TTL pulses provided by the digital pulseprogrammer. After the switching stage, the RF pulsespass into a constant-gain (+33 dBm) RF power ampli-fier. The power amplifier feeds the amplified pulsed RFinto the probe circuit.

CarrPurc.

1st pulsewidth

2nd pulsewidth

tau

RepeatTimeN

singlepair ofpulses

repeatedpairs ofpulses

Threepulse

FIG. 5.

The signal out of the sample, as well as a considerableamount of leakage during pulses, comes from the probecircuit, and is amplified by a sensitive preamp (Tron-Tech W110F). The signal then goes into a phase detector(Mini-Circuits ZRPD-1), where it is mixed with the ref-erence signal coming out of the other port of the powersplitter. Since the NMR signal is, in general, not pre-cisely at the frequency of the transmitter, when the twosignals are mixed, a signal is produced at the dierencefrequency of the resonance signal and the applied RF.Since we are looking at NMR signals in the vicinity of1-8 MHz, mixing this down to a lower frequency makesit easier to see the structure of the signal.

III.4. Digital Pulse Programmer

Most of the controls that you will manipulate are onthe digital pulse programmer, the oscilloscope or thefunction generator. The keypad of the Digital Pulse Pro-grammer is shown in the figure. Press any of the fourbuttons on the right to select a parameter (First PulseWidth (PW1), Second Pulse Width (PW2), Tau (τ ),or Repeat Time). Then use the arrow buttons to setthe corresponding time for that parameter. The defaulttimes are: PW1 = 24 s, PW2 = 48 s, τ = 2 ms, andRepeat Time = 100 ms. Note that when the repeat timeor τ is long, the pulse programmer responds slower asit needs to complete one cycle to change the settings.The top two buttons on the left determine whether atwo-pulse sequence occurs only once (the Single Pair ofPulses button), or repeats continuously (the RepeatedPairs of Pulses button) with a pause between sequencesof a length set by the Repeat Time parameter. Thethird button, labeled ”Carr-Purcell”, will create a seriesof pulses corresponding to the Carr-Purcell technique de-scribed in the Measurement section. Finally, the fourthbutton, ”Three Pulse”, outputs 180 − τ − 90 − 180

pulses for a measurement. For this pulse sequence, the90 pulse time should be set as PW1 and the 180 shouldbe PW2.

Set the delay τ , to the minimum position and observethe amplified RF pulses from the port marked ”trans-mitter” on channel 2 of the oscilloscope. The pulses

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FIG. 6.

should be approximately 20–30 volts peak-to-peak (notethat the settings on the function generator should be 2–3 volts, since there is an additional amplifier). Choosethe slowest possible sweep speed; this will enable bothpulses to be viewed simultaneously. A good starting pairof pulse-widths might be 24 s and 48 s, corresponding toapproximately 90 and 180. Now switch to channel 1,which displays the output of the phase detector (throughthe low-pass filter). Incidentally, there is another low-pass filter which is part of the scope itself. On the Tek-tronix analog scope there is a button marked ”BW limit20 MHz”, which limits the allowed bandwidth. This but-ton should be pressed in (active). On the HP digitalscope the BW limit is set by one of the soft keys. On anAgilent scope, this is set in the channel 1 or channel 2menu. Set the y-sensitivity to about 10 mV/div at first.Channel 1 will display the NMR signal. Place the glycer-ine vial in the probe and place the probe in the magnet.Now the fun begins!

Refer to the following figure, which is a highly stylizedversion of the signals you might obtain. The form of thevoltage displayed during the two bursts is unimportant.You will be focusing your attention on the FID signalsthat appear after each burst, and on the echo. For five orten microseconds after the RF pulse the amplifier is stillin the recovery phase, so this part of the signal should beignored.

III.5. The Probe Circuit

The probe circuit is a tuned LC circuit, impedancematched to 50 ohms at the resonant frequency for effi-cient power transmission to the sample. The inductorL in the circuit is the sample coil, a ten turn coil of 18copper wire wound to accommodate a standard 10mmNMR sample tube. The coil is connected to ground ateach end through tunable capacitors Cm and Ct, to al-low frequency and impedance matching. Power in andsignal out pass through the same point on the resonantcircuit, so that both the power amplifier and the sig-nal preamp have a properly matched load. Between thepower amplifier and the sample is a pair of crossed diodes,in series with the probe circuit from the point of view ofthe power amplifier. By becoming non-conducting at lowapplied voltages, these serve to isolate the probe circuitand preamp from the power amplifier between pulses,reducing the problems associated with power amplifiernoise. The crossed diodes however, will pass the high

FIG. 7.

RF voltages that arrive when the transmitter is on. Thesignal out of the probe circuit passes through a quarter-wavelength line to reach another pair of grounded crosseddiodes at the input of the preamp. The diodes short thepreamp end of the cable when the transmitter is on, caus-ing that end of the cable to act like a short circuit. Thishelps to protect the delicate preamp from the high RFpower put out by the power amplifier. Any quarter-wavetransmission line transforms impedance according to thefollowing relation:

Zin =Z20

Zout(24)

where Z0 is the characteristic impedance of the line.Therefore during the RF pulse, the preamp circuit withthe quarter-wave line looks like an open circuit to theprobe and does not load it down. Between pulses, thevoltage across the diodes is too small to turn them on,and they act like an open circuit, allowing the small NMRsignal to pass undiminished to the preamp.

IV. MEASUREMENTS

IV.1. Measurements Overview

The nature of this experiment allows for considerablevariety in the possible measurement sets which could beperformed. We present in the following sections severalwell-known techniques in pulsed NMR used to determinerelaxation times, nuclear magnetic moments, and so on.These measurement techniques are applied to varioussamples in order to construct a measurement set meetingthe basic experimental objectives of this lab. Generally,a basic set of measurement in this lab involves the fol-lowing procedures:

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1. Examine the signal chain, particularly the probe-head and the connections in the setup. If necessary,readjust the position of the probehead to place thesample in a region of more uniform/stronger mag-netic field. In the initial search for a signal, it ishelpful to experiment with 100 % glycerin and thespin echo sequence, using the two-pulse-repeatedprogram with appropriate initial guesses for PW1,PW2, and repeat time. Make notes of variousstages in the signal chain, such as the external trig-ger and the input and output signals to and fromthe probehead.

2. Using appropriate samples, determine the Larmorfrequencies for 1H and 19F by dialing the functiongenerator frequency and looking for resonance. Usea Hall effect magnetometer to measure the mag-netic field in the sample coil. Keep in mind that itmay be necessary to retune the probehead circuitand redo the search for signal when working withfluorine.

3. Determine t90, the pulse width that rotates the z -magnetization into the transverse plane. Note thatthis value can change from session to session, so itshould be reassessed each time the setup is altered.

4. Pick a set of samples on which to examine spin-spinrelaxation times. For each sample in the set, usean appropriate NMR pulse sequence to determinethe value of T2 . Look for interesting trends in T2across the sample set.

5. Pick a set of samples on which to example spin-lattice relaxation times. For each sample in theset, use an appropriate NMR pulse sequence to de-termine the value of T1. Look for interesting trendsin T1 across the sample set.

The space of possible samples that are amenable toNMR analysis is obviously enormous, but there are sev-eral samples which have traditionally been used in thislab (and which have been prepared for your use). Exceptfor the fluorine-based samples, which are used primarilyto measure the magnetic moment of the fluorine nucleus,most of these samples are based on the 1H nucleus. Thesamples used in this lab include:

• Glycerin-water mixtures: Various mixtures of glyc-erin with water, with proportions given in percent-ages by weight. Spin-spin interactions generally in-crease with liquid viscosity. Thus, measurements ofT2 as a function of glycerin ratio are of particularinterest.

• Paramagnetic ion solutions: Two ten-fold serial di-lutions of 0.830M and 0.166 M starting solutionsof Fe3+ ions. The presence of paramagnetic ionsgreatly facilitates the dissipation of energy from thespins to their surroundings. Thus, measurements of

T1 as a function of concentration are of particularinterest.

• Fluorine samples: There are samples of both triflu-oric acetic acid and hexafluorobenzene at the setup.The former is a strong acid and should be treatedwith care.

• Water: There are a number of potentially inter-esting but somewhat difficult measurements whichcould be done with water. These are discussed laterfor those interested.

IV.2. Suggested Progress Check

The optimal schedule for this lab is highly dependenton what measurement sets are planned. In general, it isa good idea to focus on utilizing one technique for eachof T1 and T2 measurements and apply that technique toa set of samples (say, glycerin for T2 and paramagneticions for T1) in order to obtain a trend. Additional samplesets and pulse sequences (or even variations on the pulsesequences) can then be added once those measurementsare complete.

Each sample set generally takes between one to two labsessions, so approximately three lab sessions should bededicated to performing relaxation measurements. Theremaining sessions should be used in the beginning to fa-miliarize yourself with the equipment and to determinethe magnetic moments of the hydrogen and fluorine nu-clei. Delays sometimes occur when signal is lost due toequipment changes or subtle changes in oscilloscope set-tings. In such cases, after obvious debugging has beendone, it is best to obtain the help of a lab technicianrather than to spend too much time tracking down aproblem.

IV.3. Finding Larmor Frequencies

The signal seen at the oscilloscope is the FID emittedby the sample at frequency γB0, mixed with a steadysignal of frequency ω from the function generator. Thelatter is the signal used to pulse the sample at near res-onance. This produces a beat signal which has the samedecay envelope as the FID but which has a comparativelylow frequency |γB0 − ω| that can easily be picked up byan oscilloscope. This is manifested on the oscilloscope asa decaying sinusoid at frequency |γB0 − ω|.

It follows that in order to determine the Larmor fre-quency, we need to tune the function generator frequencyω until the beat frequency vanishes:

|γB0 − ω| = 0. (25)

The value of ω read off from the function generatoris thus a measurement of the Larmor frequency of the

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sample magnetization. On the scope, the approach toresonance should look like a decaying sinusoid as its fre-quency goes to zero (or its period goes to infinity): theresult is simply an exponential decay.

Another way to think about this measurement is toconsider the mixing as a ”stroboscopic” view of the mag-netization vector with strobe frequency ω, used to ob-serve the precessing magnetization which has natural fre-quency γB0. When the strobe frequency matches thenatural frequency, the magnetization vector appears tostay stationary in the transverse plane, and it simply ap-pears to decay away with the time constant T ∗

2 .Once the Larmor frequency ω = γB0 has been found,

it is necessary also to measure the magnetic field seen bythe sample. This is accomplished using a Hall effect mag-netometer, which is shared by several other experimentsand so is not a part of the permanent setup. Make cer-tain to zero and calibrate the magnetometer before useand make measurements perpendicular to the magneticfield lines. Estimate the variations in the magnetic fieldover the sample.

Once the resonance frequency has been found, it isuseful to return the oscillator frequency back to beingslightly off resonance. Being able to observe a beat signalcarried by the exponential decay envelope makes identi-fying and assessing pulse sequences easier. As long asω ≈ γB0, it is still possible to perform rotations of themagnetization. Adjust ω around resonance to obtain asatisfactory FID signal.

IV.4. Finding Pulse Widths

To obtain an FID signal, it is unnecessary to use theexact value of t90 when pulsing the sample. When firstsearching for a signal or when making resonance mea-surements to find the Larmor frequency, almost any rea-sonable initial guess for PW1 will result in an observableFID signal (since α 6= 0) . However, when utilizing es-tablished NMR pulse sequences, it is important to haveaccurate values of t90 and t180 to use.

As we mentioned before, it is difficult to know themagnitude of the perturbing field B1, so the formulat90 = π/γB1 is not very helpful. However, we do knowthe amplitude of the FID should be maximum after a 90pulse and minimum or zero after a 180 pulse. Thus, oneeasy way to obtain the pulse widths is to find the settingof PW1 which minimizes the FID. This gives t180 andhalving that gives t90.

Another method is to use spin echoes. Set PW1 tosome initial value and set PW2 to be exactly twice PW1.Then adjust PW1 (keeping PW2 twice PW1) until a spinecho is visible and is maximal. Then PW1 gives t90 whilePW2 gives t180 . Obviously, there are many other waysin which the pulse widths can be obtained.

Experiment with these or other techniques in order toestimate the 90 and 180 pulse widths as closely as possi-ble. However, it is important also to realize that the pulse

programmer is only capable of setting PW1 and PW2 inunits of 1 s each. Thus, it is only profitable to narrowdown the pulse widths to within one or two microsec-onds. Generally, pulse sequences and measurements ofrelaxation times work well even if the pulse widths areslightly off. When in doubt, it may be helpful to test howbig of an effect that changes in PW1 and PW2 make onspecific pulse sequences.

Note that the values of t90 and t180 are subject tochange from one session to the next, depending on thetuning of the probehead circuit, the power output of thefunction generator, the exact placement of the sample inthe magnetic field, and so on. Thus, it is a good idea toquickly check the pulse widths every session for consis-tency.

IV.5. The 90–180 Sequence

It is evident from our discussion of the spin echo se-quence that it can be used to make measurements of spin-spin relaxation and hence T2. In this context, we call thespin echo sequence the ”90-180” pulse sequence (thedelay time is assumed). Measuring the degradation ofthe spin echo as a function of τ reveals the effect of spin-spin relaxation, as if the FID were not affected by fieldinhomogeneities.

This sequence can be configured by setting PW1 equalto t90 and PW2 equal to t180 , using the two-pulse-repeated program on the pulse programmer. The spinecho is produced a time 2τ after the initial 90 pulse.

An interesting scope technique applicable to this andother pulse sequences is the use of infinite persist. Withsuitable trigger settings, it is possible to see the spin echomoving towards the right end of the scope’s screen as weincrease τ on the pulse programmer. As the spin echomoves, its amplitude exponentially decays, and the re-sult of viewing the whole process under infinite persistis a ”decay envelope” traced out by the peak of the spinecho. It is up to each group to decide if this is an ap-proach useful for making measurements, but it is a goodvisualization nevertheless.

IV.6. The Carr-Purcell Sequence

As mentioned before, the residual loss of spin echo am-plitude after a 90-180 pulse sequence is due not onlyto spin-spin relaxation, and the spin echo suffers an ad-ditional loss in the presence of diffusion. This is partic-ularly pronounced for samples with large T2, since suchsamples require a correspondingly large τ when using the90-180 method. But diffusion effects go as τ3 in theexponential, so at large τ they dominate the spin-spinrelaxation (which go as τ), thereby causing us to under-estimate T2.

This problem was addressed also by Carr and Purcellin 1954, and they introduced a sequence (now called the

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Carr-Purcell sequence) which is much less susceptible todiffusion effects [7]. Rather than repeating a pulse se-quence with ever-increasing delay times, the Carr-Purcellmethod uses a fixed τ , which we can choose small enoughto neglect diffusion.

The Carr-Purcell method prescribes the following se-quence:

90 − τ − 180 − 2τ − 180 − 2τ − 180 − 2τ..., (26)

continuing for as long as the repeat time allotted. Thefirst two pulses are exactly the 90-180 sequence, anda spin echo is produced halfway between the first andsecond 180 pulses. But as soon as that spin echo sub-sides (after the 2τ window has passed), we again pulsethe sample with a 180, causing yet another spin echo toappear, and so on.

Thus, the Carr-Purcell method produces a train of spinechoes by repeatedly refocusing the magnetization. Witheach iteration, the spin echo amplitude decays away dueto spin-spin relaxation, but neither field inhomogeneitiesnor diffusion effects play a role in that decay (if τ ischosen small enough).

The Carr-Purcell sequence can be configured by set-ting PW1 equal to t90 and PW2 equal to t180, using theCarr-Purcell program on the pulse programmer. The re-peat time effectively determines how many iterations areapplied before the sequence repeats.

IV.7. The 90–90 Sequence

The 90-90 pulse sequence is the simplest pulse se-quence used to measure spin-lattice or T1 relaxation. Itconsists of a 90 pulse, followed by a delay of time τ , fol-lowed by another 90 pulse. The amplitude of the secondFID is then measured. It can be configured by settingPW1 and PW2 both equal to t90, using the two-pulse-repeated program on the pulse programmer.

The idea of the 90-90 pulse sequence is to first rotatea thermalized z -magnetization into the transverse plane,and then wait for some delay time τ , during which someof the longitudinal magnetization will recover via spin-lattice relaxation. The second 90 pulse then rotates thisrecovered magnetization into the transverse plane, whichgives an FID with amplitude equal to the recovery. Theunrecovered component is rotated into the 180 position,where it does nothing.

IV.8. The 180–90 Sequence

The 180- 90 pulse sequence is a variation on the9090, also used to measure T1. It consists of a 180

pulse, followed by a delay of time τ , followed by a 90

pulse. The amplitude of the FID produced by the lastpulse is measured. It can be configured by setting PW1

to t180 and PW2 to t90, using the two-pulse-repeated pro-gram on the pulse programmer.

In the 180-90 pulse sequence, the thermalized z -magnetization is flipped, inverting the spin population.Relaxation then proceeds by first reducing the magneti-zation back towards zero and then finally to equilibrium.(This pulse sequence is sometimes called an ”inversionrecovery” for this reason.) After allowing this process tooccur for a time τ , a 90 pulse is applied, bringing therecovered magnetization into the transverse plane, whereit generates an FID.

Unlike the 90-90, the magnetization in this case ac-tually reverses, going through zero at time T1 ln 2. How-ever, the amplitude of the FID is insensitive to the signof the magnetization prior to the 90 pulse (which wouldshow up as a phase shift in the sinusoid), so the FID am-plitude as seen through the scope will appear to shrinkwith small enough τ , go through zero, and then expo-nentially recover at large τ .

IV.9. The Three-Pulse Sequence

The 90–90 and 180–90 pulse sequences have thedisadvantage that they require reading the amplitude ofthe FID which occurs immediately after an RF pulse.This problem was addressed several years ago by twoJunior Lab students, who proposed a variation, the”three-pulse” sequence. (Both of these students, RahulSarpeshkar and Isaac Chuang, are now MIT professors.)

The three-pulse sequence consists of a usual inversionrecovery sequence, so the first 180 pulse inverts the ther-malized z -magnetization, and after a delay of τ , the sec-ond 90 pulse rotates the recovered z -magnetization intothe transverse plane, where it generates an FID. But in-stead of measuring the FID, we wait an additional (non-variable) time ε before applying another 180 pulse. Theeffect of this last pulse is to cause a spin echo, whose am-plitude reflects the amplitude of the FID. But since thespin echo is separated in time from the RF burst, it ismuch easier to measure. The amplitude of the spin echoafter the third pulse as a function of τ thus follows thesame trend as in the 180−90 sequence.

The three-pulse sequence can be configured by settingPW1 to t90 and PW2 to t190, and then using the three-pulse program on the pulse programmer. The secondtime delay εcannot be set manually, and has been pro-grammed to be about 1 ms. The time ε is kept small tominimize spin-spin effects that might occur.

IV.10. Relaxation in Water

In describing the various pulse sequences, we assumedfor the most part that the two relaxation mechanismsare not simultaneously important. For example, whenmeasuring T2 , we assume that a negligible amount of

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the spin echo decay is due to actual recovery of the z -magnetization via spin-lattice relaxation. For the mostpart, this assumption is valid when working with samplesin which one relaxation constant is drastically smallerthan the other (e.g., T2 << T1 in viscous liquids andT1 << T2 in paramagnetic solutions).

Of course, whether this assumption is valid for anyparticular sample is something which deserves considera-tion in analyzing your results. In fact, this assumption isnot quite true for many samples used in NMR, where T1and T2 are usually comparable. An important example iswater, where both of the relaxation constants are on theorder of several seconds, making measurements of bothquite difficult.

There is, however, at least one interesting way to mea-sure T1 in water which deserves mentioning. As discussedpreviously, if we want each pulse sequence to yield an in-dependent measurement, we must set the repeat time onthe pulse programmer sufficiently large to allow rether-malization. For example, executing two spin echo se-quences too close together will make the second spin echoappear smaller. This suggests that we can actually takeadvantage of this fact to make a measurement of T1 us-ing spin echoes and by varying not the delay time butthe repeat time.

Use the standard spin-echo sequence (90–180) withsome fixed time constant τ . Record the spin echo heightproduced when repeating the sequence using a variablerepeat time; the spin echo height as a function of therepeat time should give the exponential return of themagnetization with time constant T1. For higher repeattimes higher than 3 s, you can use the two-pulse-singleprogram on the pulse programmer and a watch, ratherthan the two-pulse-repeated program. As an optionalexperiment, perform these measurements with both tapwater and distilled water. Is it possible to detect thedifference?

The first measurements of T1 in distilled water stoodfor about thirty years. Since then, careful measurementshave produced a number which is about 50% higher. Thedifference is due to the effect of dissolved oxygen in thewater, as O2 is paramagnetic. As an optional experiment,try to remove the dissolved oxygen from a sample of dis-tilled water and see if there is any difference. One waythis could be done is by bubbling pure nitrogen through

the water, as N2 is diamagnetic.

V. ANALYSIS

Due to the wide range of possible measurements inthis experiment, there is a corresponding variety in theparticular analysis approach that best suits your data set.However, some results that are often presented includethe following:

• The Larmor frequencies and gyromagnetic ratios ofthe proton and 19F nucleus.• Demonstration of the successful use of a pulse se-

quence to measure the spin-lattice relaxation timeT1 across a range of samples (e.g., paramagneticion solutions).

• Demonstration of the successful use of a pulse se-quence to measure the spin-lattice relaxation timeT2 across a range of samples (e.g., paramagneticion solutions).

The idea is to structure the measurement sets and anal-ysis so that you can make a case for the effectiveness ofNMR as a way of probing the microscopic structure ofyour samples, by demonstrating measurements that aresensitive to changes in the material structure. You mayfind throughout the experiment that some samples areeasier to work with than others. Time constraints canalso dictate which measurement sets to use and whatanalysis approach to take.

One interesting idea often pursued is to reproduce theearly results by Bloembergen and Purcell when workingwith water-glycerin mixtures, such as those in [2] andBloembergen’s thesis [8]. They found that plotting therelaxation constants against logarithm of viscosity (orconcentration) resulted in interesting curves as the vis-cocity (concentration) varied over a large range. Tablesrelating percent weight of glycerin to viscosity can befound in tables like this one collected by DOW.

Remember as always to address sources of errors oruncertainty in your results, quantitatively whenever pos-sible. Each pulse sequence is susceptible to differentsources of systematic effects (e.g., diffusion in the 90180

sequence), so interpretation of the results needs to takeinto account an understanding of the physics behind thetechnique. In some ways, this one of the central ideasbehind NMR spectroscopy.

[1] F. Bloch, Phys. Rev. 70, 460 (1946).[2] N. Bloembergen, E. Purcell, and R. Pound, Phys. Rev.

73, 679 (1948).[3] F. Bloch, Nobel Lecture (1952).[4] E. M. Purcell, Nobel Lecture (1952).

[5] A. Abragam, Principles of Nuclear Magnetism (Ox-ford University Press, 1961) physics Department ReadingRoom.

[6] M. Levitt, Spin Dynamics: Basics of Nuclear MagneticResonance.

[7] H. Carr and E. Purcell, Phys. Rev. 94, 630 (1954).

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[8] N. Bloembergen, Nuclear Magnetic Relaxation (W.A.Benjamin, 1961) physics Department Reading Room.

Appendix A: Quantum Mechanical Description ofNMR

Recall that for all spin-1/2 particles (protons, neu-trons, electrons, quarks, leptons), there are just twoeigenstates: spin up |S, Sz〉 = | 12 ,

12 〉 ≡ |0〉 and spin down

|S, Sz〉 = | 12 ,−12 〉 ≡ |1〉. Using these as basis vectors, the

general state of a spin-1/2 particle can be expressed as atwo-element column matrix called a spinor :

|ψ〉 = u|0〉+ d|1〉 =

[ud

]. (A1)

Normalization imposes the constraint |u|2 + |d|2 = 1.The system is governed by the Schrodinger equation

i~d

dt|ψ〉 = H|ψ〉, (A2)

which has the solution |ψ(t)〉 = U |ψ(0)〉, where U =e−iHt/~ is unitary. In pulsed NMR, the Hamiltonian

H = −~µ · ~B = −µ [σxBx + σyBy + σzBz] (A3)

is the potential energy of a magnetic moment placed inan external magnetic field. The σi are the Pauli spinmatrices:

σx ≡[

0 11 0

],

σy ≡[

0 −ii 0

],

σz ≡[

1 00 −1

]. (A4)

Inserting Equations (A4), (A1), and (A3) into Equa-tion (A2), we get

u = (µ/~) [iBx +By] d+ i(µ/~)Bzu,

d = (µ/~) [iBx −By]u− i(µ/~)Bzd. (A5)

If Bx = By = 0 and the equations reduce to

u = i(µ/~)Bzu,

d = −i(µ/~)Bzd. (A6)

Integrating with respect to time yields

u =u0ei(µ/~)Bzt = u0e

iω0t/2,

d =d0e−i(µ/~)Bzt = d0e

−iω0t/2, (A7)

where ω0 = 2µBz/~ is the Larmor precession frequency.If an atom undergoes a spin-flip transition from the spinup state to the spin down state, the emitted photon hasenergy E = ω0~.

Now let’s add a small external magnetic field Bx butstill keeping By = 0 and such that Bx Bz. Equations(A5) become

u =i(µ/~)Bxd+ i(µ/~)Bzu,

d =i(µ/~)Bxu− i(µ/~)Bzd. (A8)

For a time varying magnetic field of the type producedby an RF burst as in pulsed NMR, Bx = 2Bx0 cosωt =Bx0

(eiωt + e−iωt

). Define ωx ≡ 2µBx0/~. We see that

u =i(ω0/2)u+ i(ωx/2)(eiωt + e−iωt

)d,

d =− i(ω0/2)d+ i(ωx/2)(eiωt + e−iωt

)u. (A9)

Using ωx ω0 (since Bx B0), we can try for asolution of the form

u =Cu(t)eiω0t/2,

d =Cd(t)e−iω0t/2. (A10)

Inserting Equations (A10) into the differential equations(A9) for u and d, we get

Cu =iωx2Cd

[ei(ω−ω0)t + e−i(ω+ω0)t

],

Cd =iωx2Cu

[ei(ω+ω0)t + e−i(ω−ω0)t

]. (A11)

If the system is run near resonance (ω ≈ ω0), then Equa-tion (A11) becomes

Cu =iωx2Cd,

Cd =iωx2Cu. (A12)

We have also used ω0 ωx. Taking the derivatives ofthese equations, we see that Cu and Cd act like harmonicoscillators of frequency ωx/2. These have the generalsolution

Cu =a cos (ωxt/2) + b sin (ωxt/2),

Cd =ia sin (ωxt/2)− ib cos (ωxt/2). (A13)

Putting these in Equation (A10), we get the solution foru and d. These are called Rabi oscillations, valid forωx ω0.

Appendix B: Bloch Sphere Representation

A single qubit in the state u|0〉+d|1〉 can be visualizedas a point (θ, φ) on the unit sphere, where u = cos(θ/2),d = eiφ sin(θ/2), and u can be taken to be real becausethe overall phase of the state is unobservable. This iscalled the Bloch sphere representation, and the vector(cosφ sin θ, sinφ sin θ, cos θ) is called the Bloch vector.

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The Pauli matrices give rise to three useful classes ofunitary matrices when they are exponentiated, the rota-tion operators about the x, y, and z axes, defined by theequations:

Rx(θ) ≡e−iθσx/2 = cosθ

2I− i sin

θ

2σx

=

[cos θ2 −i sin θ

2

−i sin θ2 cos θ2

], (B1)

Ry(θ) ≡e−iθσy/2 = cosθ

2I− i sin

θ

2σy

=

[cos θ2 − sin θ

2

sin θ2 cos θ2

], (B2)

Rz(θ) ≡e−iθσz/2 = cosθ

2I− i sin

θ

2σz

=

[e−iθ/2 0

0 eiθ/2

]. (B3)

One reason why the Rn(θ) operators are referred to asrotation operators is the following fact. Suppose a single

qubit has a state represented by the Bloch vector ~λ. Thenthe effect of the rotation Rn(θ) on the state is to rotateit by an angle θ about the n axis of the Bloch sphere.

An arbitrary unitary operator on a single qubit canbe written in many ways as a combination of rotations,together with global phase shifts on the qubit. A usefultheorem to remember is the following: Suppose U is aunitary operation on a single qubit. Then there existreal numbers α, β, γ and δ such that

U = eiαRx(β)Ry(γ)Rx(δ). (B4)

Appendix C: Fundamental equations of magneticresonance

The magnetic interaction of a classical electromagneticfield with a two-state spin is described by the Hamil-

tonian H = −~µ · ~B, where ~µ is the spin, and B =B0z+B1(x cosωt+ y sinωt) is a typical applied magneticfield. B0 is static and very large, and B1 is usually timevarying and several orders of magnitude smaller than B0

in strength, so that perturbation theory is traditionallyemployed to study this system. However, the Schrodingerequation for this system can be solved straightforwardlywithout perturbation theory. The Hamiltonian can bewritten as

H =~ω0

2Z + ~g(X cosωt+ Y sinωt), (C1)

where g is related to the strength of the B1 field, and ω0

to B0, and X, Y , and Z are introduced as a shorthandfor the Pauli matrices. Define |φ(t)〉 = eiωtZ/2|χ(t)〉, suchthat the Schrodinger equation

i~∂t|χ(t)〉 = H|χ(t)〉 (C2)

can be re-expressed as

i~∂t|φ(t)〉 =

[eiωZt/2He−iωZt/2 − ~ω

2Z

]|φ(t)〉. (C3)

Since

eiωZt/2Xe−iωZt/2 = (X cosωt− Y sinωt), (C4)

Equation (C3) simplifies to become

i∂t|φ(t)〉 =

[ω0 − ω

2Z + gX

]|φ(t)〉, (C5)

where the terms on the right multiplying the state can beidentified as the effective “rotating frame” Hamiltonian.The solution to this equation is

|φ(t)〉 = ei

[ω0−ω

2 Z+gX

]t|φ(0)〉. (C6)

The concept of resonance arises from the behavior ofthis solution, which can be understood to be a singlequbit rotation about the axis

n =z + 2g

ω0−ω x√1 +

(2g

ω0−ω

)2 (C7)

by an angle

|~n| = t

√(ω0 − ω

2

)2

+ g2. (C8)

When ω is far from ω0, the spin is negligibly affectedby the B1 field; the axis of its rotation is nearly parallelwith z, and its time evolution is nearly exactly that of thefree B0 Hamiltonian. On the other hand, when ω0 ≈ ω,the B0 contribution becomes negligible, and a small B1

field can cause large changes in the state, correspondingto rotations about the x axis. The enormous effect asmall perturbation can have on the spin system, whentuned to the appropriate frequency, is responsible for the‘resonance’ in nuclear magnetic resonance.

In general, when ω = ω0, the single spin rotating frameHamiltonian can be written as

H = g1(t)X + g2(t)Y, (C9)

where g1 and g2 are functions of the applied transverseRF fields.

Appendix D: Modeling the NMR Probe

The material in this appendix was provided by Profes-sor Isaac Chuang. A tuned circuit is typically used toefficiently irradiate a sample with electromagnetic fieldsin the radiofrequency of microwave regime. This circuitallows power to be transferred from a source with min-imal reflection, while at the same time creating a largeelectric of magnetic field around the sample, which istypically placed within a coil that is part of it.

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FIG. 8. Schematatic diagram of a typical NMR probe circuit.The connector on the right goes off to the source and anydetection circuitry.

1. Circuit and Input Impedance

A typical probe circuit, as shown in Figure 9, consistsof an inductor L, its parasitic coil resistance R, a tuningcapacitor CT , and an impedance matching capacitor Cm.We can analyze the behavior of this circuit using themethod of complex impedances, in which the capacitorshave impedance ZC = 1/iωC, inductors ZL = iωL, andresistors ZR = R, with ω = 2πf being the frequency inrad/sec. The input impedance is thus

Z =ZCm+

[1

ZCT

+1

R+ ZL

]−1

=1

iωCm+

[iωCT +

1

R+ iωL

]−1

=1 + iωR(CT + Cm)− ω2L(CT + Cm)

iωCm(1 + iRωCT − ω2LCT ). (D1)

2. Tune and Match Conditions

The resonant frequency of this circuit is set by

ω2∗ =

1

L(CT + Cm), (D2)

and at this frequency, the input impedance is

Z0 =R(CT + Cm)

Cm(1 + iRω∗CT − ω2∗LCT )

. (D3)

We would like this impedance to be 50 Ω, because thatis the typical impedance expected by RF or microwavesources and the coaxial cable which carries in the signal.Setting Z0 = 50 we obtain:

50 Ω

R=

(CT + Cm)2

Cm [Cm + iRω∗CT (CT + Cm)]. (D4)

To good approximation, the iRω∗CT (CT + Cm) term inthe denominator may be neglected, giving

50 Ω

R=

(1 +

CTCm

)2

. (D5)

When these conditions are satisfied, almost all thesource power goes into the tuned resonator at the res-onant frequency, thus creating the strongest possible os-cillating magnetic field inside the coil L.