Nuclear Magnetic Resonance

Giovanni Pinuellas

13 December 2010

Abstract

In this lab we observe nuclear magetic resonance phenomenon in glyc-erin, water and teflon. We performed two experiments. The continuouswave NMR experiment was used to determine the resonance frequency ofour samples as well as the magnitude of the magnetic field of our perma-nent magnet. The pulsed NMR expeiment was used to observe spin echophenomenon and determine the relaxation time T2.

1 General TheoryNuclear magnetic resonance (NMR) is the phenomenon that happens when mag-netic nuclei are immersed in a static magnetic field and then exposed to anotheroscillating magnetic field. Magneic atomic nuclei have a magnetic moment, µ,that is proportional to the angular momentum and a constant of proportionalityknown as the gyromagnetic ratio, γ. Usually we consider a frame of referencealong the z-axis (the static field is orientied along the z-axis), so that

µ = γSz

where Sz is z-component of the angular momentum vector S. For spin 1/2 par-ticles (such as the protons we will be observing in this experiment) Sz = m� =± 1

2�, since m takes the values +1/2 (spin down) or -1/2 (spin up). At a zeroexternal field the energies of these magentic moments are degenerate, but whenplaced in a magnetic field these two states decouple. The interaction betweenthe magnetic moment in a magnetic field results in a magnetic interaction knownas the Zeeman interaction. The interaction energy of the magnetic dipole µ ina magnetic field Bo is defined by the hamiltonian

H = −µ ·Bo = −γm� ·Bo

and since we orient Bo in the z-direction,

H = −µBo = −γm�Bo

and therefore the separation of energies between Zeeman splittings is alwaysγ�Bo for whatever m.

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In the Schrödinger picture

i� ∂

∂t|Ψ(t) >= H|Ψ(t) >

and if H is contant with time then the Schrödinger equation yields the result

|Ψ(t) >= U(t)|Ψ(0) >

whereU(t) = exp (−iHt/�) = exp (iγmBot)

is the evolution operator, which is identical to a clockwise rotation of the stateabout the z-axis by an angle γBot. This has the effect that all states will precessat the Larmor frequency ωo given by

υo = γBo.

This precession is called Larmor precession, and is anologous to the classicalpicture as a magnetized compass needle experiencing a torque by its intercationbetween the north and south poles of a magnet. The magnetic field of thismagnet will interact with the compass needle (the magnetic moment of thenucleus) to create equal and opposite forces at its ends that creates the torque

τ = γM×B

and thus begings to rotate. The solution to this equation when B =Boz alsocorresponds to a Larmor precession of the magnetization about the field at therate υo = γBo.

We can define an ensemble magnetization M that is the defined as the netmagnetization of individual magnetic moments. In this model the macroscopicangular momentum vector is M/γ, and equating the torque to the rate of changeof angular momentum we have

dM

dt= γM×B. (1)

In the equilibrium state the ensemble magnetization lies along the directionof the static field Bo. We denote this equilibrium magnetization as Mo. Thez-component of M is called the longitudinal magnetization, Mz, and in thisconfiguration M =Mz = Mo. The transverse magnetizations here, Mx and My

are 0.We apply an external RF field to disturb the spin state from its thermal

equilibrium state. This has the effect of tilting M to the transverse plane byexposing the spin states to an energy of a frequency equal to the energy differ-ence between the states. In time this equilibrium is restored by a spin-latticerelaxation process, which involves the exchange of energy between the spins andthe surrounding thermal reservoir (the ’lattice’) which it is in equilibrium. Thisprocess is governed by the equation

dMz

dt= −Mz −Mo

T1(2)

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with solutionMz = Mo(1− 2 exp (−t/T1)). (3)

Here T1 is called the spin-lattice (or longitudinal) relaxation time. It is thetime constant which describes how Mz returns to its equilibrium value Mo, asseen in (3). As mentioned, it is mainly due to the thermal interactions of the spinstates. Additionally, with the introduction of paramagnetic ions such as M++

the T1 time is significantly decreased as the stronger magnetic moments restorethe ensemble magnetization to its equilibrium state faster. This is because thewidth of the voltage response versus time plot has a 1/T1 relationship.

We can apply a “π/2” pulse to our spin system to completely shift the ensem-ble magnetization M to the xy plane (transverse magnetization). This rotatesthe magnetization 90º. With M = Mx +My, the transverse magnetization willbegin to precess around the z-axis with frequency υo. But here the inevitableinhomogeneity of the applied magnetic field over the dimensions of the ensem-ble will disrupt the Larmor frequencies of all the spins. This will essentiallycause the spins to dephase as individual spins will precess at different rates,with differing rates. Eventually in time the spins can completely decay to yieldMxy = 0. This decay is called the called spin-spin relaxation and it is describedby

dMx,y

dt=

−Mx,y

T2(4)

with solutionMx,y = Mx,y(0) exp (−t/T2) (5)

where Mx,y(0) is the initial transverse magnetization immediately after the π/2pulse is applied.

Spin-spin relaxation is governed by T2, the spin-spin relaxation time. T2

is the time constant that describes the transverse magnetization returning tothermal equilibrium among themselves. It is also called the dephasing time,which is the time required for Mx,y to decay to 0. For T2 processes, spin-latticeeffects are still present, but the effect of phase decoherence between nuclear spinstates dominates. This results in T2 ≤ T1.

It is convinient to define a rotating plane S� which rotates about the longi-

tudinal axis at the Larmor frequency υo. The Z axis remains unchanged but wedistinguish the rotating X and Y axes by X’ and Y’. Correcting (1) using (2),and (4) in this rotating frame yields the set of relationships known as the Blochequations:

dMx

dt= γ(M×B)x − Mx

T2

dMy

dt= γ(M×B)y −

My

T2

dMz

dt= γ(M×B)z +

Mo −Mz

T1.

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Without damping (none of the T1 or T2 terms) the Bloch equations are justas in (1):

dMx,y,z

dt= γ(M×B)x,y,z.

Magnetization can be measured by free nuclear induction created by theprecession of the ensemble magnetization. Surrounding our sample is a receivercoil that when precession occurs, a voltage is induced in the coil. This voltageis defined by V = −dφ

dt , or the negative of the flux through the coil. Dependingon the degree saturation, we can observe either an adiabatic (slow passage)or a non-adiabatic (fast passage) resonance pattern based on the amount ofparamagnetic ions in our samples. Adiabatic passage corresponds to a smallerT1, while non-adiabatic passage corresponds to a bigger T1.

We can measure this induced voltage with our electronics. Additionally, wecan meausure the spin-spin relaxation time T2 by reproducing a process knownas spin echo. After an RF pulse of π/2 is applied, the transverse magnetizationdephases. Here we can say that the spins in frame S

� rotating with Larmorfrequency υ = υo + u have precessed by an angle ut. We can then apply anadditional π pulse which corresponds to a rotation of 180º about X

� of theplane S

�. At the end of the pulse the spins of frequency υo+u will have a phase−u(t + τ) with respect to X

� and thus at a time 2(t + τ) they will be alignedagain. Therefore the transverse magnetization will add constructively to thevalue equal to the one immediately following the first pulse.

Physically this π/2 and π pulse procedure has the effect of reversing all spinprecessions. This results in the spins recohering as the slower spins are now infront and the faster ones in the back. The faster precessions will recombine torephase to Mx,y(0), then dephase again to 0. Half of this time is equivalent tothe spin-spin relaxation time T2.

2 The Continous Wave NMR Experiment

2.1 Apparatus and TheoryWe first investigate NMR with the continous wave (CW) NMR experiment. Herewe examine protons and F19 in a static magnetic field provided by a permanentmagnet of Bo ≈ 3.9 kG (with the purpose of measuring the magnetic fieldBo as precicely as we can with this procedure). We observe NMR phenomenaby using a continuous RF field tuned to resonance and modulating the staticfield sinusoidally around resonance. This method probes the induced voltageresponse at a fixed resonance with varying magnetic field. Alternatively, wecould have left the magnetic field fixed and varied the RF frequency.

We apply the RF field B1 perpendicularly to the static field. This field isprovided by a tunable RF oscillator coil. The frequency is determined by aninductor and a capacitor, where υo = 1/

√LC. One variable capacitor tunes the

frequency. This coil in enclosed in a black box we call the NMR head. The RFfield can be set to either absorption or dispersion by adjusting a copper disk

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on the cavity of the NMR head. This changes the direction of B1 relative tothe receiver coil, or in other words, the phase of the leakage voltage into thereceiver coil. With this we can view an absorption resonance pattern where wemodulate across the magnetic field at almost an angle of the field B1 perpen-dicular to the direction of the static field Bo. The resonance pattern should bevery symmetrical about Bo, as we see in their respective figures. In dispersion,because this angle is off, the leakage voltage is dependant on the frequency andthus we see a dispersion pattern. The amplitude of B1 is controlled by the DCsupply voltage to the oscillator.

For the samples we use glycerin, H2O+MnCl24H2O at a range of molarities,and F19 in the form of teflon rods. We place the tubes inside the NMR headwhere they are wrapped around receiver coils to pick up the induced voltagefrom the precessing ensemble moment. The NMR head is placed between thestatic magnetic field of a permament magnet. This static field is modulated at60Hz by an oscillating field Bmod by two small coils located on the top and thebottom of the NMR head.

With the RF field tuned to the Larmor frequency corresponding to υo = γBo,the RF signal is absorbed by the spins and it produces the precessing ensemblemoment cause by the shifting to the transverse plane. This precession inducesan emf in the receiver coils, which when plotted against the input modulationresults in the resonance patterns in an oscilloscope or with a LabView programon the computer.

Further investigation involved fine-tuning the NMR resonance by leavingthe RF frequency constant and applying another small field B2 by using awave generator to drive another set of wire coils. We use a lock-in amplifierto sweep slowly across a range of magnetic fields. We set this modulationto be very small and record the derivative of the output signal. The lock-inamplifier is used to detect and measure very small AC signals by a phase-sensitive detection technique that singles out the component of the signal at avery specific reference frequency and phase. Noise signals at frequencies otherthan the reference frequency (here 60 Hz) are ignored. This is accomplished bymuliplying the signal by a pure sine wave at the reference frequency. Becausesine waves of different frequencies are orthagonal, when integrating over a periodall other signals are filtered. The output is recorded in a strip chart using theLabView program.

2.2 ResultsFor glycerin we find the resonance frequency to be within the range of 16.525±0.001MHz. This error is due to the systematic error due to the accuracy of the fre-quency counter. This yields a magnetic field Bo = 3881.8±0.0046 gauss. Thefollowing is the calibrated resonance pattern for glycerin non-adiabatic (fast)passage absorption. This results because the concentration of paramagneticions is low. We can clearly see characteristc “wiggles” which indicate the inho-mogeneity of the magnetic field. We can estimate the inhomogeneity from thedamping rate of these wiggles, which we detremine to be 1.50±0.25 gauss.

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We replace the glycerin sample with a 0.01M Mn++ solution in water. Herethe signal is also that of non-adiabatic absorption, where we can also clearly seethe characteristic wiggles.

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With the 1M Mn++ sample we observe both absorption and dispersion res-onance patterns, but this time adiabatically (slow passage). This is because ofits high concentration of paramagnetic ions decreased T1, which corresponds toa slow passage. The two plots in figure 1 show these patterns.

We also observed the resonance patterns of T19 nuclei in a teflon rod. Herewe measured a resonance of 15.538±0.001 MHz. Here the absorption peak isless pronounced than the protons’ because of the larger mass of the flourinenucleus. The patterns of both absorption and dispersion are shown in figure 2.Measuring both resonance frequencies of H1 and F19 under identical conditionswe can determine the ratio of their resonance frequencies. We find that theratio of F19 to H1 is 0.940±9.451×10−6. This is also the ratio of their magneticmoments.

We then observe sequence of absorption lock-in trace from the computer formolarities M=0.03, 0.1, 0.3, 1, 3.3 Molar Mn++. These are shown in figures 3,4, 5, 6 and 7, respectively. As we can observe, the line widths slightly decreasewith increased molarity. This is to be expected, as an increased concentration ofprotons will decrease the T1 relaxation time as the stronger magnetic momentsrestore the ensemble magnetization to its equilibrium state faster.

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Figure 1: Resonance patterns for 1M++

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Figure 2: Resonance patterns for F19

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Figure 3: Strip chart for 0.03M Mn++

Figure 4: Strip chart for 0.1M Mn++

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Figure 5: Strip chart for 0.3M Mn++

Figure 6: Strip chart for 1M Mn++

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Figure 7: Strip chart for 3.3M Mn++

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3 The Pulsed NMR Experiment

3.1 Apparatus and TheoryIn The CW NMR experiment we observed nuclear resonance of the ensemblemagnetization of our samples in a static magnetic field by modulating this fieldat 60 Hz while applying a continuous RF field perpendicular to the static field.The alternative in the pulsed NMR experiment is to instead apply a short intenseRF pulse and observe the signal after the pulse is removed. This has the sameeffect as like the ringing of a bell. We will observe the ringing down of the signalafter the pulse is applied.

We apply an RF pulse at near the Larmor precession of the spins using a CWRF oscillator modulated by a DG535 pulse generator. The duration of the pulseis calibrated by the pulse generator to correspond to a 90º (π/2) rotation. Thisis achieved by setting the digital delay settings on the DG535 to the time thatit takes to translate the ensemble magnetization from the longitudinal planeto the transverse plane, which in turn is figured out by searching for the timethat corresponds to the maximum voltage response in our detector coil due tonuclear induction by the precessing spins.

Once they’re down in the transverse plane we turn off the excitation pulse.The spins precess at the Larmor frequency but after a time the spins will dephasebecause of field inhomogeneities. This is the free induction decay. This causesthe signal to decay as the spins dephase to form a “pancake” that makes Mxy = 0.This dephasing can be reversed by applying a π pulse that reverses the spinsand recoheres Mxy. This pulse is applied just by extending the duration ofthe second pulse to double the time of the first pulse. Since the first pulsecorresponded to a π/2 pulse, this pulse will correctly correspond to a π pulse.

If we apply our second pulse at time t then the echo will occur at time 2t. T2

can be measured by the spin echo response. Since this spin echo recoheres thendecays again with a maximum amplitude identical to the first free inductiondecay response, half the time of the duration of the spin echo is equal to thespin-spin relaxation (or dephase) time T2.

The NMR head for this experiment is replaced by a smaller one with asingle receiver coil. This head is placed inbetween the permanent magnet linedup at the center of the magner where the static field is most homogeneous. TheRF pulses to excite the system are produced by the CW RF oscillator whichproduces continous wave RF voltage at close to the Larmor frequency. Wegate this CW RF oscillator into brief pulses by a pulse genetator and a mixer.The output of the mixer is a choped section of the square wave outputed bythe pulse generator and the RF voltage. This pulse is then amplified by anRF power amplifier and then sent to matching capacitors and then to an RFcoil that is used to excite the spins. The precessing magnetization is detectedby the receiver coils and sent through another amplifier and then to a digitaloscilloscope and to the computer where we can actually analyze our signals.

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Figure 8: Spin echo effect of glycerin

3.2 ResultsWe found that for glycerin the best duration for a π/2 pulse is 80µs and 3mslater a π pulse of 160µs. Free induction decay is seen only when we are a little offthe resonance frequency. When we are at the resonance frequency we only seethe envelope of the decay. The next figure shows the spin echo effect on glycerin.The large spikes are the RF pulses overloading our detectors and their responsesare immediately after. We see our first pulse occur at the 5ms mark followedby the free induction decay. The second pulse is applied 3ms later at 8ms andthe spin echo is observed at 11ms. Our estimation of T2 is 0.30ms±0.02ms.

4 ConclusionTraditionally T2 is also measured by applying a Fourier transform to the spinecho. But our Fourier transform was incoherent because we took our signalexactly at resonance, producing no wiggles.

Additionally, T1 could have been measured by applying three succesive π/2pulses. We first apply a π/2 pulse that rotates the magnetization to the trans-verse plane. Another π/2 rotattion rotates the decayed signal back to the Z

axis, then another π/2 pulse is applied to measure magnetization at time t. Ifwe measure Mz at different t we can use (3) to calculate T1. Unfortunately this

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cannot be measured as we can only apply two RF pulses at a time.Other interesting measurements could be to measure the decay of the spin

echo. In the spin echo procedure we can continue to reapply π pulses to con-tinue to recohere the phases of the spins. Due to other inhomogeneities in theenvironment we will find that the transverse magnetization will not recohere toa magnitude idential to the last one. This decay of spin echo signals also cannotbe measured by the current NMR setup.

5 References[1] Paul T. Callaghan. “Principles of Nuclear Magnetic Resonance Microscopy.”1991 Oxford Science Publications.[2] A. Abragam. “Principles of Nuclear Magnetism.” 1961 Oxford UniversityPress.[3] Felix Bloch. “Nuclear Magnetism.” American Scientists: Vol. 43, No. 1, Jan1955, pp. 48-62.[4] Felix Bloch. "Nuclear Induction.” Physical Review 70, 460. No 7 and 8,(1946).

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