Download Pulsed Nuclear Magnetic Resonance and Spin Echo

(Advanced) Experimental Physics
V85.0112/G85.2075
Fall, 2013
August 15, 2013
Pulsed Nuclear Magnetic Resonance and Spin Echo
Tycho Sleator
Goals
The goal of this experiment is to explore various types of coherent transient phenomena involving the interaction of proton magnetic moments in water with radio-frequency magnetic fields.
Background
This experiment is an example of magnetic resonance - a class of experiments in which very small energy
separations (typically 10−7 eV) of quantum mechanical states are determined by measuring the frequency
of a sinusoidally varying magnetic field, which induces transitions between the states. Some examples of
magnetic resonance experiments are: nuclear magnetic resonance [1, 8], nuclear quadrupole resonance [8]
electron spin resonance [9], atomic beam magnetic resonance [10], and optical pumping [11]. These different
experiments are distinguished by the systems under study i.e. atoms, nuclei, or solids, and by the method of
detection of the resonance. Magnetic resonance has played an important role in almost every field of physics.
Examples of the physical quantities measured in these experiments are given below.
Field of Research
Elementary Particles
Nuclear Physics
Solid State Physics
Atomic Physics
Biology
Medicine
Examples of Quantities Measured by Magnetic Resonance
Magnetic moments of the electron, proton, and neutron. Structure of
positronium and muonium. Upper limit on the electric dipole moment
of the neutron
Nuclear magnetic dipole moments, electric quadrupole moments, magnetic
octupole moments, and spins
Spin-lattice and spin-spin relaxation times. Knight shifts. Magnetic fields
at the nucleus for ferromagnetic and anti-ferromagnetic solids. Energy level
structure of paramagnetic ions in crystals and the structure of VK centers
Fine and hyperfine structures. Zeeman and Stark effects. Lifetimes of
excited states. Lamb shifts.
Structure of organic molecules and proteins (NMR), study of photosynthysis (EPR)
Magnetic resonance imaging
In addition, through magnetic resonance it has become possible to measure magnetic fields to better than a
part in 106 and to construct new time standards based on atomic resonance frequencies (The cesium atomic
clock is presently used).
Nuclear Magnetic Resonance in Diamagnetic Substances
Interaction of Nuclear Spins with a Static Magnetic Field
For a diamagnetic substance, such as water, the local magnetic field at the nuclei is only slightly lower (by
a few parts per million) than an applied field. The nuclei interact with the applied magnetic field because
of their magnetic moments. This is exactly analogous to the interaction of atomic magnetic moments with
the external magnetic field, which gives rise to the Zeeman effect in atomic spectroscopy. The interaction
energy is given by E = −µ · H, where H is the applied magnetic field. The magnetic moment vector µ is
Pulsed Nuclear Magnetic Resonance and Spin Echo
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parallel to the spin angular momentum vector J = ~I of the nucleus (where I is a dimensionless spin angular
momentum operator). The relationship between the two vectors is conventionally written as:
µ = γ~I = gµN I,
(1)
where γ is called the gyromagnetic ratio, µN ≡ e~/2Mp c is the nuclear magneton, Mp is the mass of the
proton and g is a dimensionless nuclear g-factor, which for protons is about g = 5.59. If we apply a static
magnetic field along the z-direction, H0 = H0 ẑ, the Hamiltonian for a given nucleus can be written
H0 = −γ~H0 · I = −γ~H0 Iz ,
(2)
where Iz is the z-component of I. The possible energies of the system are given by the eigenvalues of the
Hamiltonian:
Em = −γ~H0 m.
(3)
The quantum number m is an eigenvalue of Iz with possible values I, I − 1, I − 2, · · · , −I, where I is the
quantum number for the total angular momentum of the nucleus. For the case of protons (in e.g., water),
where I = 1/2 and g > 0, the energy level difference between states m = ±1/2 is given by ∆E = γ~H0 , as
shown in Figure 1. This energy difference can also be expressed in terms of a frequency ω0 = ∆E/~ = γH0 .
Figure 1: Energy levels of a spin 1/2 nucleus. The corresponding frequency is ω0 = ∆E/~ = γH0
Interaction with an oscillating field along x
In the experiment, a magnetic field H1 (t) = 2H1 cos ωt x̂ oscillating at frequency ω, nearly resonant with the
transition frequency ω0 , is applied along the x-direction, as shown in Figure 2. In the case of protons (used
Figure 2: Magnetic field geometry
Pulsed Nuclear Magnetic Resonance and Spin Echo
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in this experiment) with I = 1/2, this field introduces a time-dependent interaction term in the Hamiltonian
HI = ~γH1 (t)Ix = ~γ 2H1 cos ωt σx ,
(4)
where σx is a Pauli spin matrix representing the x-component of the spin of a spin-1/2 particle. If the
resonance condition ω ∼ ω0 , is approximately satisfied, then transitions will occur between the m = ±1/2
states. More details of the interaction of the nuclear spins with an rf (radio frequency) magnetic field will
be given after a discussion of thermal equlibrium.
Thermal Equlibrium
Experimentally, we measure the frequency f = γH0 /2π. The detection of the resonance (equality of f
applied and γH0 /2π) is made possible by the fact that there is a slight excess of nuclei in the lower energy
states. Because of this excess the application of an oscillating (rf) magnetic field of the correct frequency
can lead to (1) absorption of energy from the circuit producing this field and (2) other transient effects due
to the motion of the total magnetic moment of the sample.
If the populations of the m = ±1/2 states are exactly equal, any transitions between these states are
unobservable, since they are equally probable in either direction. For there to be an observable effect of the
H1 field, there must be an initial population difference between the m = ±1/2 states. Let N+ and N− denote
the populations of the m = +1/2 and m = −1/2 states, respectively. A fundamental property of systems in
thermal equilibrium at absolute temperature T is that the populations N± of the various states of energy
Em are proportional to e−Em /kB T , where kB is Boltzmann’s constant, and T is the absolute temperature.
Thus, for thermal equilibrium at temperature T , the ratio
N−
= e−∆E/kB T ,
N+
(5)
where ∆E ≡ E+1/2 − E−1/2 . One can show from Eq. (5) that the population difference n ≡ N+ − N− is
given by
∆E
n = N tanh
(6)
2kB T
∆E
,
(7)
≈ N
2kB T
where N ≡ N+ + N− is the total population of protons, and the last equation (7) applies when kB T ∆E,
which is always the case for nuclei at room temperature.
An important physical quantity describing the system is the magnetization of the sample. The magnetization M is the magnetic moment per unit volume. In the case of thermal equilibrium, where individual
magnetic moments can be described as either pointing along the field or opposite the magnetic field (assumed
to point along the z-direction), we can write Mz = nµ = nγ~/2, where µ = γ~/2 is the magnetic moment of
a given spin. We therefore have (in the high temperature approximation) for the equilibrium magnetization
of the sample
γ~
γ~ ∆E
γ~ ~γH0
(~γ)2 H0
µ2 H0
M0 = nµ = n
=N
=N
=N
=N
(8)
2
2 2kB T
2 2kB T
4kB T
kB T
Note that the last equality on the right of Eq. (8) indicates that the equilibrium magnetization increases
with the value of H0 . This increased magnetization results in a larger signal, motivating the use of very high
fields (generating by superconducting magnets) of close to 20 T and operating at frequencies near 800 MHz.
The rate at which thermal equilibrium is approached (in the absence of any rf field) is characterized by
the “spin-lattice relaxation time” T1 , and is modeled by the following equation:
dMz
M0 − Mz
=
,
dt
T1
where M0 is the value of Mz at thermal equilibrium.
(9)
Pulsed Nuclear Magnetic Resonance and Spin Echo
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Behavior of nuclear spins in the presence of static and oscillating magnetic fields
Pression about a static field H0
Consider a particle with magnetic dipole moment µ and angular momentum J , such that µ = γJ . In the
presence of a magnetic field H0 = H0 ẑ, we find that there is a torque on the dipole given by τ = µ × H0 .
Since this torque gives the rate of change of the angular momentum, we have
dµ
= γµ × H0 .
dt
(10)
Since H0 points along ẑ, we find that
µ̇x
=
γH0 µy
µ̇y
=
−γH0 µx
µ̇z
=
0
(11)
and the magnetic moment precesses about H0 (maintaining a constant angle between µ and H0 ) with
frequency ω0 = γH0 . The direction of precession is clockwise when viewed from the positive z axis.
Rotating frame of reference
Now consider a frame of reference that is rotating at angular velocity ω clockwise as viewed from the positive
z-axis. From this frame of reference (called the rotating frame), and in the presence of only the magnetic
field H0 , the spins precess at an angular frequency γH0 − ω = ω0 − ω. This is equivalent to the spins
precessing in an effective field Heff = ẑ(ω0 − ω)/γ. The rotating frame becomes particularly useful if we
want to consider the behavior of the spins in the presence of an additional magnetic field H1 (t) that is
precessing about H0 at frequency ω. In a typical NMR experiment, H1 will be turned on for brief period of
time (a few µs). If we transform to the rotating frame of reference at this frequency ω, then the field H1 is
stationary in this frame (pointing, say, along the x0 axis in the rotating frame). In the rotating frame, we
then have the spins precessing about a static magnetic field, whose z-component is given by (H0 − ω/γ)ẑ,
and whose x-component is H1 x̂0 . This situation is depicted in Figure 3.
Figure 3: Precessing spins as viewed in the rotating frame of reference. The spins precess about the direction
of the effective field Heff , which is given by Heff = (H0 − ω/γ)ẑ + H1 x̂0 . The circle indicates the path of the
precessing magnetic moment, and the little arrows give the direction of precession.
Under the conditions of exact resonance, in which ω = ω0 , (or ω = γH0 ), the effective field is given by
H1 x̂0 and points along the x0 axis in the rotating frame, as shown in Figure 4. If the system is in a state of
thermal equilibrium before H1 is turned on, then the initial direction of M will be along the z-axis. During
the pulse, the magnetization vector M precesses about the direction of H1 , and remains confined to the
Pulsed Nuclear Magnetic Resonance and Spin Echo
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Figure 4: Resonant “θ” pulse. When ω = γH0 , there is no ẑ component to the effective field Heff , so
Heff points along the x0 -axis in the rotating frame. During the pulse, the magnetization (indicated by the
magnetic moment µ) rotates by an angle θ about the x0 -axis.
z-y 0 plane in the rotating frame. During the pulse, the magnetization will therefore rotate about the x0 -axis
by an angle θp = γH1 tp , where tp is the duration of the pulse. If θp = π/2, then after the pulse is turned
off the magnetization lies in the x0 -y 0 plane. Such a pulse is referred to as a π/2-pulse, or a 90◦ -pulse. If
θp = π, then after the pulse, the magnetization points along the −z-axis. A π pulse therefore has the effect
of inverting the initial magnetization.
How H1 is produced
Up until this point it hasn’t been explained how the rotating field H1 is produced. Notice that in Figures
2 and 5, the magnetic field along the x-axis is 2H1 cos ωt. This field can be decomposed into two separate
a)
b)
Figure 5: Magnetic field geometry: view from the positive z-axis. a) The linear polarized magnetic field
2H1 cos ωt x̂ can be written as the sum of two counter-rotating magnetic fields, each with magnitude H1 .
The counter-rotating field H01 (t) has very little effect on the spins, and therefore can be neglected. The
rotating field H1 lies along the x0 -axis in the rotating frame. b) Adding a phase shift φ to the magnetic field
results in the magnetic field H1 being rotated by an angle −φ from the x0 -axis in the rotating frame.
fields, each with magnitude H1 , rotating in opposite directions in the x-y plane, as shown in Figure 5a).
That is, H01 + H1 = 2H1 cos ωt x̂. One of these fields H01 is rotating in the opposite direction from the
natural precession of the spins in the field H0 , and therefore has negligible effect. The other field H1 is
directed along the x0 -axis in the rotating frame, and is responsible for rotating the spins about the x0 -axis,
as described above, and shown in Figure 4.
Pulsed Nuclear Magnetic Resonance and Spin Echo
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By changing the phase φ of the rf magnetic field applied to the sample, so that H(t) = 2H1 cos(ωt+φ), as
shown in Fig. 5b), one can change the direction that H1 points in the rotating frame. In the case of a phase
shift of φ, the direction of H1 is rotated clockwise as viewed from the positive z-axis by an angle φ. The
ability to change this angle is useful in certain pulse sequences, such as the “Carr-Purcell-Meiboom-Gill”
(CPMG) sequence. This pulse sequence (which will discussed in more detail later) uses magnetic field pulses
that are oriented both along the x0 and y 0 -axes in the rotating frame.
Bloch equations
In 1946, Felix Bloch developed a set of phenomenological equations that describe the behavior of the spins
in an NMR experiment [1]. These equations take into account both the precession of the spins about the
applied magnetic field, as well as relaxation effects that are due to interaction of the nuclear spins with each
other and with the environment. The Bloch equations are
dMx (t)
dt
dMy (t)
dt
dMz (t)
dt
=
=
=
Mx (t)
T2
My (t)
γ(M(t) × H(t))y −
T2
Mz (t) − M0
γ(M(t) × H(t))z −
T1
γ(M(t) × H(t))x −
(12)
(13)
(14)
The first term on the right of each equation just reflects the fact that dM/dt = γM × H, a result that follows
immediately from Eq. (10). The last term on the right of Eq. (14) comes from the tendency of the spins to
become polarized along the magnetic field, as indicated by Eq. (9). The last term in equations (12) and (13)
represent the decay of the transverse magnetization (that is, the components of the magnetization transverse
to the direction of the magnetic field H0 ). This decay is a consequence of the interaction of each spin with
neighboring spins. The neighboring spins of a given spin modify slightly the magnetic field seen by the
given spin, and therefore affect the rate of precession of that spin. Since different spins see slightly different
magnetic fields (from the neighboring spins), the spins precess at slightly different rates and consequently
get out of phase with each other, resulting in a reduction of the transverse magnetization. The time scale
for the reduction of this magnetization is referred to as T2 . One thing you will do in this experiment is to
make measurements of T1 and T2 of the sample.
There is one more important effect in the experiment that has to do with the fact that the field created
by the magnet is not perfectly homogeneous over the volume of the sample. In this case, once the spins are
flipped down into the x-y plane, they precess at different rates by virtue of the fact that the field produced
by the magnet, H0 , is different as seen by different spins. If H0 is sufficiently inhomogeneous (which it is in
this experiment), then the precessing spins will get out of phase with each other over a time that is much
shorter than T2 . This shorter time scale is referred to as T2∗ . Thus, we have, in general:
T1 > T2 > T2∗ .
(15)
In this experiment you will make measurements of these quantities.
Free induction decay: Measurement of T2∗
Simply by applying a π/2 pulse to the spins, and recording the duration of the resulting signal from the
precessing spins allows one to measure T2∗ .
How to measure T1
A measurement of T1 can be made by the following technique. After waiting a long time (much longer
than T1 ), apply a π pulse to the spins. This will invert the spins, so that there is an excess number of
spins pointing opposite the magnetic field direction. Over time, the spins will evolve toward the eqilibrium
value as described by Eq. (9). So after the initial π pulse one waits a variable time τ before applying a π/2
pulse, and observing the size of the resulting FID. The size of the FID will be proportional to the size of
the magnetization just before the π/2 pulse. A plot of the FID amplitude as a function of τ should be an
Pulsed Nuclear Magnetic Resonance and Spin Echo
7
exponential curve with time constant given by T1 . Note that the experiment needs to be repeated a number
of times with different values of τ , and that the time interval between repetitions should be several times
the value of T1 .
How to measure T2 : Spin echo
The value of T2 can be measured by performing a series of spin-echo experiments [3]. The principle behind
the spin-echo experiment is shown in Figure 6. In a spin-echo experiment, a π/2 pulse is applied, followed
a)
b)
c)
d)
Figure 6: Spin Echo. a) π/2 pulse along the x0 -axis in the rotating frame flips the magnetization M onto
the y 0 -axis; b) Because the magnetic field is inhomogeneous, different spins precess at different rates. The
fastest precessing spins, labeled with an “f” are precessing to the right in the rotating frame, and the slowest
precessing spins (“s”) are precessing to the left; c) At time τ , the application of a π pulse (along the x0 -axis)
rotates the spins about the x0 -axis by 180◦ . The spins continue to precess at the same speed (and direction)
as before (indicated by the arrows); d) at time 2τ all the spins converge along the −x0 -axis, resulting in a
spin-echo.
after a time τ , by a π pulse. A free-induction decay is observed immediately after the initial π/2 pulse, and
also possibly after the π pulse. In addition, an echo signal is observed at a time 2τ after the initial π/2 pulse.
The explanation for this effect is given in Figure 6, and the caption.
Alternative method to measure T2 : Carr-Purcell and Carr-Purcell-Meiboom-Gill (CPMG)
sequence
One drawback of the previous technique for measuring T2 is that after each repetition of the experiment, one
has to wait several T1 times, increasing the time it takes to collect the necessary data. (Before any repetition
of an experiment, one should wait for the spins to come into thermal equilibrium.) Another disadvantage is
that if the magnetic field is not completely homogeneous, then diffusion of the proton spins can contribute
to a reduced echo lifetime. The Carr-Purcell sequence [5] allows one to determine the value of T2 in a single
measurement, and can minimize the error caused by diffusion. In this experiment, at time t = 0 an initial
π/2 pulse is applied, and at times t = τ , 3τ , 5τ , etc, a π pulse is applied. This results in a series of echoes at
times 2τ , 4τ , 6τ , etc. The height of each successive echo as a function of the time of that echo allows one to
determine the value of T2 . See if you can figure out why this works by extending the diagrams in Figure 6.
There is one drawback to the Carr-Purcell sequence in that if the “π” pulse is not exactly a π pulse, then
successive π pulses will tip the spins further and further out of the x0 -y 0 plane, and will contribute to the
decay of the signal. The solution to this is the Carr-Purcell-Meiboom-Gill (CPMG) sequence, in which the
π pulses are applied along the y 0 -axis in the rotating frame, instead than the x0 -axis. In this case (see if you
can see why), errors in a given π pulse will be mostly cancelled out by the succeeding π pulse.
Pulsed Nuclear Magnetic Resonance and Spin Echo
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Measurement of diffusion
Since we are mostly interested in liquid samples, there is the possibility that the spins in the sample can
diffuse to a new position during the experiment. If the magnetic field H0 is homogeneous across the sample,
then this diffusive motion is not detectable. However, by the application of a gradient in H0 across the
sample, the motion of a given spin will result in a frequency shift, and the diffusive motion of the spins can
be detected by the reduction in the echo-lifetime as measured by a two-pulse Hahn echo sequence.
A spin echo is based on the fact that the total precession angle between the first two pulses is equal to
the total precession angle between the second pulse and the echo. [Even though the rate of precession may
be different for different spins, they still end up pointing in the same direction at the time of the echo. See
Figure 6.]
In the presence of a magnetic field gradient, a spin that moves around (diffuses) will have a precession
frequency that varies with time (as the spin moves to areas where the magnetic field is different). The result
is that the total precession angle between the two pulses won’t be equal to the precession angle between the
second pulse and the echo. So, if you consider a large number of spins, they in general won’t be pointing in
the same direction at the time of the echo, and the size of the echo will be smaller then the case where there
is no diffusion.
A detailed calculation yields the following result for the magnetization:
M (t) = M0 exp(−γ 2 G2 Dt3 /12),
(16)
where G is the magnetic field gradient and D is the diffusion constant [5, 6]. So, by measuring the echo
lifetime in the presence of a known magnetic field gradient, one can determine the diffusion constant of the
spins (i.e., the water molecules) in the sample.
Apparatus
Overview
An overview of the experimental setup is shown in Fig. 7. In this experiment, a set of radio frequency (rf)
pulses is applied to the sample coil and the resulting precessing magnetic moment is picked up by the same
coil. The radio frequency is generated by an oscillator, and switched on and off by an rf-switch controlled
by a pulse generator. These pulses are then amplified by a 5 W power amplifier before going to the probe
circuit.
The probe circuit consists of the pickup coil and a pair of adjustable capacitors. It is configured in such
a way that with appropriate adjustment of the capacitors, the impedance of the circuit when “looking into
it” is 50 Ω. By matching the impedance of the probe circuit to the characteristic impedance of the coaxial
cables carrying the signal, and the other components in the circuit (amplifiers, etc), one can insure that all
of the power generated by the power amplifier is transfered to the probe circuit, and any signal generated
by the probe circuit is efficiently transfered to the detection circuitry.
The precessing magnetic moment induced by the applied pulses generates a voltage (emf) across the
pickup coil, which is amplified by the low noise amplifier before being “mixed” with the signal from the
oscillator. This “mixed-down” signal, which is at a frequency equal to the difference between the oscillator
frequency and the frequency of the precessing spins is filtered and amplified further before being digitized
and saved to a file.
Details of the Circuit
rf Signals, Coaxial Cables, and Impedance Matching: To fully understand the operation of the
apparatus, it is important to understand a few things about radio frequency electronics. One important
distinction between radio frequency signals and audio (low frequency) signals, is that the distances between
components in a radiofrequency circuit can be a significant fraction of the wavelength. Remember that the
wavelength λ of a signal is related to the frequency f by λ = v/f , where v is the speed of the signal (equal
to the speed of light c when the signal propagates through a vacuum). For a frequency of f = 20 MHz, the
wavelength is 15 m.
Pulsed Nuclear Magnetic Resonance and Spin Echo
9
iSpin-NMR
Controller
Reference
Oscillator
Transmitter
S
0
Receiver
S
90
0
90
Low - Pass
Filter
Low - Pass
Filter
Pulse
Generator
S
Power
Amp.
S
Preamp
λ/4
=
Probe tuning circuit
Sample
Coil
Figure 7: Functional diagram of the apparatus: Large dashed box is the iSpin-NMR controller, the two
dash-dot boxes inside the controller labeled “Transmitter” and “Receiver” delineate the transmitter and
receiver parts of the controller, and the small dashed box is the probe-tuning circuit. f0 = spectrometer
frequency, fs = spin precession frequency, s = power splitter/combiner, 0◦ = 0◦ phase shifter, 90◦ = 90◦
phase shifter, ⊗ = Mixer, —
×– = crossed diodes, λ/4 = quarter-wavelength of coaxial cable. Note that there
is an additional preamp outside of the iSpin controller not shown in this diagram (see Fig. 8).
In this experiment, co-axial cables are used to convey rf signals. These cables have a characteristic
impedance of 50 Ω, which is the ratio of the voltage to the current at any point along the cable when a
traveling wave is propagating through it. The velocity of a signal through the cable is typically about (2/3)c.
When a device is connected to the end of the cable some fraction of the signal will be reflected off the device
back through the cable, depending on the impedance of the device. If the device has an impedance of 50
Ω (i.e., if its impedance is matched to the cable), then there will be no reflection and all the power in the
signal will be absorbed by the device. In general (but not always), when dealing with rf-signals, one uses 50
Ω coax cables and devices with 50 Ω input and output impedances.
Pulsed Nuclear Magnetic Resonance and Spin Echo
10
Power Splitters: If one wants to run an rf signal to two components, one often uses a power splitter. The
purpose of using the power splitter is to split the signal while maintaining an impedance of 50 Ω. A power
splitter can also be used to combine signals from two separate cables into a single cable. Power splitters are
labeled by the symbol “S” in Figure 7.
Low-Pass Filter: This device simply absorbs (or reflects) any frequencies in the signal that are above the
“cut-off frequency”. The only signals passing through the device are those with frequency below the cut-off
frequency.
Mixers: A Mixer is a 3 port device (it has 3 connectors). When signals are sent into two of the ports, the
mixer effectively multiplies these signals together and sends the product out the third port. As an example,
suppose that two signals of frequencies ω1 and ω2 are entering two of the input ports respectively (remember
that ω = 2πf is the “angular” frequency). Assume that these signals can be described by cos(ω1 t) and
cos(ω2 t). The output of the mixer will have a signal proportional to cos(ω1 t) cos(ω2 t). It can be shown using
trigonometric identities that
cos(ω1 t) cos(ω2 t) = cos[(ω1 + ω2 )t]/2 + cos[(ω1 − ω2 )t]/2.
(17)
Thus, a mixer produces a signal consisting of frequencies equal to the sum and difference of the frequencies
at the inputs. Often the sum frequency is eliminated using a low-pass filter, resulting in only the difference
frequency. Mixers used for this purpose are shown by the symbol ⊗ in the receiver section of Figure 7.
A mixer can also be used as a rf switch. If an rf-signal is presented at one input and a dc (constant
voltage) signal is presented at the other input, the output will consist of an rf signal whose amplitude is
proportional to the size of the dc signal. In particular, if the dc signal has zero voltage, then no rf will exit
the mixer. By turning on and off the dc voltage, one can switch the rf signal on and off. Mixers used for
this purpose are shown by the same symbol (⊗) in the transmitter section of Figure 7.
Mixers are used for two purposes in this experiment. They are used to produce rf pulses by combining
the signals from the rf-oscillator and a pulse from the pulse generator (see the transmitter section of Fig. 7).
A mixer is also used to reduce the frequency of the precessing nuclear spins (something close to 21 MHz) to
a frequency equal to the difference between the precession frequency and the oscillator frequency (see the
receiver section of Fig. 7). The mixer also produces the sum frequency, but this frequency is removed by the
low-pass filter. More details of how this is done are given in Appendix I.
Crossed Diodes: Crossed diodes consists of 2 diodes in parallel, but pointing in opposite directions (see
Fig. 7). A diode has the property that current can only pass through the diode in one direction (in the
direction of the arrow on the diagram). For purposes of this discussion, a diode, when forward biased
(voltage at the tail of the arrow larger than the voltage at the tip), has the properties that no current will
pass through the diode if the voltage across the diode is less than about 0.6 V, and that if current is passing
through the diode, the voltage across the diode is 0.6 V. For the purposes of this experiment, the crossed
diodes can be considered to behave in the following way. If the magnitude of the voltage at one side of the
pair of crossed diodes is significantly less than 0.6 V, then the crossed diodes act like an open circuit (the
signal cannot pass through, and the impedance as viewed by the incoming signal is infinite). If the incoming
signal is much larger than 0.6 V, then the crossed diodes act simply like a wire with essentially no attenuation
of the signal. One function of the crossed diodes shown in Fig. 7 is the following (another function will be
described below): The crossed diodes at the input of the preamplifier insure that during the rf pulses, the
voltage at the input of the preamplifier never exceeds 0.6 V, thus protecting the preamplifier from being
damaged by the large amplitude rf pulse. The crossed diodes at the output of the power amplifier prevents
any noise from the amplifier from getting into the detection circuit when the rf pulse is off, but allows the
rf pulse itself to get through with very little attenuation.
λ/4 coaxial cable: A length of coaxial cable that is a quarter of a wavelength long (λ/4) has the following
interesting property. If a device with impedance Zoutput is connected to one end of the cable, then the
Pulsed Nuclear Magnetic Resonance and Spin Echo
11
impedance looking into the other end of the cable Zinput will be given by
Zinput =
Z02
Zoutput
,
(18)
where Z0 = 50 Ω is the characteristic impedance of the cable. Thus, if one end of the cable is shorted
(Zoutput = 0), one will see infinite impedance looking into the other end. In the experiment, the λ/4 line
is used in the following way: During the pulse, the pair of crossed diodes at the input of the preamplifier
effectively shorts the amplifier input to ground. This end of the λ/4 cable therefore has zero impedance.
The other end of the λ/4 cable therefore has infinite impedance, and has no influence on the signal from the
power amplifier (it’s as if the λ/4 cable were not connected). Thus, all the power from the power amplifier
is directed toward the sample coil. After the pulse is turned off and the nuclear spins are producing a signal,
that signal sees an impedance of 50 Ω looking into the λ/4 cable, and infinite impedance looking into the
crossed diodes at the output of the power amplifier. The signal therefore travels down the λ/4 line and is
efficiently coupled into the preamplifier.
General Operating Procedures
Check the circuit:
Verify that the circuit is connected as shown in Fig. 8. The “iSpin-NMR Controller”
Computer
USB
iSpin-NMR Controller
RF Power Out
NMR Signal In
=
Probe tuning circuit
Sample
Coil
Figure 8: Overview of experimental setup: —
×– = crossed diodes, f0 = spectrometer frequency, fs = spin
precession frequency, “Preamp” = preamplifier.
is the black box sitting on the bench top, and contains the components shown schematically in the large
rectangular box in Fig. 7. The dashed rectangle labeled “probe tuning circuit” corresponds to two tunable
capacitors mounted on a circuit board attached to the pickup coil inside the magnet. The blue coaxial cable
going into the magnet is connected to the probe tuning circuit.
Turn on the magnet controller The magnet controller is the wooden box with an aluminum front panel
shown in Fig. 9. To turn on the magnet controller, do the following steps:
1. Turn on the power switch on the magnet power supply, which is a large grey box to the right of the
magnet controller. (Note that the switch is where the power cable enters the power supply.)
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869=;/;69$562$.$<"/$65$"G1"2;0"9/<$/7./$2"H3;2"$0.G;030$5;"E=$</.:;E;/4&$
Pulsed Nuclear Magnetic Resonance and Spin Echo
12
%C&$I"/$5"2260.>9"/;8$J</355K$5.EE$;9/6$/7"$0.>9"/$>.1$<387$.<$1.1"2$8E;1<L$1;9<L$<0.EE$/66E<L$"/8&$
$
$
$$$$$$$$$
$$$$$$$$M;>32"$C&N$$$O69/26E$?.9"E$
Figure 9: Magnet Controller: Potentiometers on the left control the magnet temperature (and frequency),
and the potentiometers on the right control the magnetic field gradients.
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$
;9="1"9="9/$/7"20.E$<"2#6$39;/<L$69"$562$".87$16E"&$$P7"<"$<"2#6$.2"$="<;>9"=$/6$A""1$/7"$
/"01"2./32"$65$/7"$;9=;#;=3.E$0.>9"/;8$=;<8$69$".87$16E"$./$.$869</.9/$/"01"2./32"&$$Q6/"$/7./$/7"$
2. Make sure the two toggle switches on the magnet controller with the labels “OPEN” and “CLOSED”
<1"8;5;8$/"01"2./32"$65$.9$;9=;#;=3.E$16E"$1;"8"$;<$96/$;0162/.9/&$$S7./$;<$"<<"9/;.E$;<$/7./$/7"$
are in the “OPEN” position.
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$
3. As you turn the “RIGHT” and “LEFT” potentiometers next to these toggle switches, you will see
the “ERROR SIGNAL INDICATORS” turn either green or red, depending on the direction you turn
P6$.88601E;<7$/7;<$</.:;E;/4L$.$<1"8;.E$/7"20;</62$/"01"2./32"$<"9<62$;<$13/$;9$>66=$/7"20.E$869/.8/$
them. Position the two potentiometers at about half way between the points where the read and green
@;/7$".87$Q=M"R$=;<8&$$D$/7"20;</62$;<$.$<1"8;.E$0./"2;.E$@76<"$2"<;</.98"$#.2;"<$;9$.$12"=;8/.:E"$
indicators turn on, and at the same value as each other. This reading on the potentiometers will tell
.9=$2"126=38;:E"[email protected]$@;/7$/"01"2./32"&$$P73<L$/7"$#.E3"$65$/7"$/7"20;</62$2"<;</.98"$;<$.$0".<32"$
you the approximate value of the Larmor (spin precession) frequency. Lock the potentiometer positions
65$/7"$/"01"2./32"$65$/7"$Q=M"R$=;<8L$@7;87$;<$.E<6$.$0".<32"$65$;/<$0.>9"/;T./;69&$$$P7"$
in place with the small lock tab on each potentiometer knob.
/"01"2./32"$65$/7"$=;<8$;<$/7"9$06=3E./"=$U869/26EE"=V$:4$.$/7"206"E"8/2;8$866E"2L$@7;87$7.<$/7"$
4. Flip the “FEEDBACK LOOP” toggle switches from the “OPEN” position to the “CLOSED” position.
2"0.2A.:E"$1261"2/4$65$";/7"2$866E;9>$62$7"./;9>$.$<325.8"$="1"9=;9>$69$/7"$16E.2;/4$65$/7"$8322"9/$
5. Make sure gradient potentiometers are set to the following values:
/7263>7$;/&$$W"./$;<$/2.9<5"22"=$5260$/7"$Q=M"R$0.>9"/;8$/6$/7"$7"./$<;9A<$@7;87$"G87.9>"$7"./$/6$
/7"$2660$:4$:6/7$869#"8/;69$.9=$2.=;./;69&$$P7"<"$7"./$<;9A<$.E<6$0.<H3"2.="$.<$/7"$<;="$1.9"E$
{X, Y, Z, Z 2 } = {+0.15, +0.60, −0.10, −2.35},
(19)
E.:"E<$65$/7"$39;/&$$B/3="9/<$<763E=$96/$13/$/7";2$7.9=<$69$/7"<"$7"./$<;9A<$=32;9>$82;/;8.E$239<$@7"2"$
where the +/− signs indicate that the toggle switch below the potentiometer should be placed in the
61/;030$/"01"2./32"$</.:;E;/4$;<$2"H3;2"=&$B60"$8.2"$<763E=$:"$/.A"9$@;/7$/7"<"$1.9"E<&$$I"//;9>$
“+/−” position. These values ensure that the magnetic field has reasonably good homogeneity across
7".#4$6:X"8/<$7;/$/7"0$863E=$=.0.>"$/7"$/7"206"E"8/2;8$866E"2&$
the sample, resulting in a long-duration FID (free-induction decay), or a narrow spectral width. It’s
$
possible that the potentiometer values that result in the best homogeneity will vary with environmental
Y1"2./;9>$/7"$<"2#6$E661$/"01"2./32"$869/26EE"2$;<$</2.;>7/[email protected]=&$$P7"$E.</$</3="9/$@76$.<$3<"=$
conditions (e.g., room temperature), so for situations where you want the best homogeneity, you can
/7"$39;/$<763E=$7.#"$E"5/$/7"$E661$Y?ZQ[$$P7;<$;<$".<;E4$=69"$@;/7$/7"$/@6$/6>>E"$<@;/87"<$69$/7"$
try adjusting these values (slightly) to produce a narrower spectral width. Record the final values of
5269/$1.9"E&$$+/$8.9$.E<6$:"$.88601E;<7"=$:4$/329;9>$/7"$16@"2$655L$3<;9>$/7"$16@"2$<@;/87$69$/7"$
the gradient potentiometers in your lab notebook.
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6. It will take about an hour for the magnet temperature to reach steady-state, but meanwhile, you can
12.8/;8"$:"8.3<"$;/$@;EE$.E<6$/329$655$/7"$"E"8/2;8$5;"E=$<7;0$>2.=;"9/$86;E<&$$+5$463$.2"$3<;9>$/7"$<7;0$
load the sample and observe a free-induction decay (FID).
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Load the sample Place the sample into the probe coil (between the poles of the magnet). Start with a
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sample of water with 10 mM copper sulfate (CuSO4 ) (labeled near the top of the sample tube). Note that
/7./$/7"$<1"8/260"/"2$.E<6$:"$E"5/$69&$$W6@"#"2L$@"$2"8600"9=$/7./$:6/7$/7"$D$.9=$R$13E<"<$69$/7"$
1 mM = 10−3 Molar, where 1 Molar is defined as 1 mole of solute per liter of solution. For proper position
13E<"$126>2.00"2$:"$/329"=$655&$
of the sample in the magnet, the bottom of the rubber o-ring near the top of the sample tube should be 39
$ mm above the center of the fluid in the bottom of the tube.
$
Observe a free-induction decay using the LabView interface Double click on the LabView in++$,$%-$ interface. This is shown in Fig. 10. Near the
$ botterface icon in the desktop to launch the LabView
tom of the window you will see a box labeled “Spin API DLL path”. Edit this path so that it reads:
“C\SpinCore\SpinAPI64\dll\spinapi64.dll”. Whenever you restart the LabView interface, you will need
to modify the path as just described.
Pulsed Nuclear Magnetic Resonance and Spin Echo
13
To observe a free-induction decay, set the parameters in the LabView interface with the values shown in
Table 1, by first clicking the tab titled “Single Pulse Parameters” near top right of the LabView window.
Since the magnetic field of the magnet depends on the magnet temperature, the “Spectrometer Frequency”
setting can be estimated from Fig. 11 by reading the value of the 10-turn potentiometers on the left side of
the TeachSpin PS2 magnet controller, and using the diagonal line in Fig. 11 to find the frequency.
Parameter
Number of Points
Spectrometer Frequency (MHz)
Spectral Width (kHz)
Pulse Time (µs)
Transient Time (µs)
Repetition Delay (s)
Number of Scans
Amplitude
Output Phase
Value
8192
see Fig. 11
100
5
150
0.5
8
0.2
doesn’t matter
Description
Number of complex valued data points to collect
Frequency of rf pulse sent to the sample
Bandwidth of the recorded data (points/ms)
Duration of the pulse
Wait time before response of spins is recorded
Time interval between repetitions of the experiment
Number of repetitions to average
Relative amplitude of pulse
Table 1: Parameter values for observing an FID signal using the SpinCore LabView Graphical User Interface.
Then press “Start Acquire” near the lower right of the window. When the number of “Scan Count” (near
the lower left of the window) reaches the value you set for “Number of Scans”, press “Download Data”. You
should see a graph of the free-induction decay (amplitude vs time) on the screen. You can repeat this exercise
with different values of the parameters. For example, try changing the Spectrometer frequency by a few
kHz and see what you get. The oscillations you see on the screen are at a frequency equal to the difference
Figure 10: LabView interface for communicating with the iSpin NMR system. Graph shows an FID from
a single pulse (upper curve) as well as the magnitude of the frequency spectrum (lower curve). The mouse
can be used to change the scale of these graphs.
Pulsed Nuclear Magnetic Resonance and Spin Echo
14
Frequency HMHzL
21.18
21.16
21.14
21.12
21.10
5.0
5.5
6.0
6.5
7.0
Potentiometer Setting
7.5
8.0
Figure 11: Approximate frequency of TeachSpin magnet as a function of the potentiometer settings.
between the spectrometer frequency and the spin precession frequency. By clicking on the “Real + FFT”
tab above the graph, you will see both the real part of the signal (as a function of time), and the magnitude
of the Fourier transform (FFT), giving the frequency spectrum of the signal.
To save data to the computer, first enter the path and file name in the “File Name” box near the bottom
of the LabView interface window, and then click the check box labeled “Output File”. Then when the
Download button is pressed, the data will be saved to the file you selected, as well as updated on the screen.
Save a few FID signals for slightly different values of the spectrometer frequency, and plot the saved files
(with the extension ‘.txt’) using the python program plot_ispin_nmr_data_FFT.py. To better see the
spectrum in the lower plot, you may want to increase the value of the parameter sf in the python program
from 1 to a larger number, like 10.
Experiments
Observation of the “Free Induction Decay” (FID)
The signal you observe on the screen, as described in the previous section, is a “Free-Induction Decay” (FID).
Study the behavior of this signal as a function of various experimental parameters, such as spectrometer
frequency, pulse duration (or “pulse time”). Make qualitative observations and record some signals onto the
computer. Attempt to explain your observations. You may need to consult some of the references (e.g., [4]).
What is responsible for the decay of the FID signal?
There is useful feature of the interface that allows you to find the resonant frequency of the spins. Click
on the “Calculations” tab near the upper right of the interface, and then click “Find Resonant Frequency”.
This will put the spectrometer on resonance with the spin precession frequency, and display the value of the
frequency found (but unfortunately, to an insufficient number of significant digits).
Optimum π/2 and π pulse durations
Use the LabView NMR interface (see Fig. 13) to determine the size of the FID signal as a function of the
duration of the pulse. Keeping the parameters the same as in Table 1, click on the “Calculations” tab, and
enter the following parameters: Initial PW = 1, Final PW = 30, Step Size = 1. Then click on the button
that says “Find 90◦ Pulse Width”. What do you observe qualitatively? What is going on physically that
produces these results? Identify the pulse durations that correspond to π/2, π, and 3π/2 pulses. Take a
screen shot and save it for further reference.
To get a more accurate value for the π/2 and π pulses, repeat the above experiment with the parameters:
Initial PW = 3 (µs), Final PW = 10, Step Size = 0.25. Note that the value of the π pulse can be determined
more precisely, because the slope of the curve (signal vs. pulse duration) is a maximum at the zero crossing
Pulsed Nuclear Magnetic Resonance and Spin Echo
15
of the signal. If you have trouble determining the π/2-pulse duration, use half the value of the π-pulse
duration. Save a screen shot of the results.
Observation of the “spin-echo”
One can observe a spin echo by clicking the “CPMG Parameters” tab in the LabView interface and choosing
the value of Necho = 1. Doing this produces a 90◦ pulse followed time τ later by a 180◦ pulse. One should
see an FID after each pulse plus a “spin-echo”, which occurs a time τ after the second pulse.
If the pulses are too close together (value of τ too small), you won’t see an echo that is separated from the
FID after the second pulse. In this case, increase the delay between the two pulses so that it is significantly
longer than the duration of the FID.
Now record the signal for several different pulse separations. Be sure to collect a data for a value of τ
that results in an echo that is significantly reduced in size. The value of τ for which the echo is about half
the FID signal is an estimate of the spin-spin relaxation time T2 . In the next section an easier, more reliable
method of measuring T2 is described.
CPMG Sequence for measurement of T2
As discussed above, there is a faster and more accurate, way to find the value of T2 by using the CPMG
sequence. One can make this measurement by a single sequence of pulses (as opposed to a set of pulse
sequences where the spacing between pulses is varied from one sequence to another). This can be done
using the “CPMG Parameters” tab in the LabView interface with the value of Necho > 1. Instead of setting
Necho = 1 as described in the previous section, set Necho to 10 or 20. For now, set the other parameters to
the following values:
Parameter
Spectrometer Frequency (MHz)
Spectral Width (kHz)
Pulse Time (µs)
Transient Time (µs)
Repetition Delay (s)
Number of Scans
Amplitude
Value
see Fig. 11
30
π/2 duration
150
0.5
8
0.2
Description
Frequency of rf pulse sent to the sample
Bandwidth of the recorded data (points/ms)
Duration of the pulse
Wait time before response of spins is recorded
Time interval between repetitions of the experiment
Number of repetitions to average
Relative amplitude of pulse
Table 2: Parameter values for observing a CPMG signal using the SpinCore LabView Graphical User Interface.
Note that there are some constraints on the parameter values. It turns out that the total number of
(complex) data points that can be acquired is ntot = 214 = 16, 384. The total duration of the CPMG scan is
(2Ne + 1) τ , and the rate of data acquisition (in data points/s) is SW. So the total number of points acquired
is SW (2Ne + 1) τ . Thus, we require that
SW (2Ne + 1) τ < 16, 384.
(20)
Exceeding this limit will result in incomplete data or an error.
The CPMG scan has an additional feature that makes makes the data easier to analyze. You can ask
the program to acquire a limited number of points for each echo. For example, if you ask the program to
acquire 5 points/echo, you will just get data from the top of each echo. The resulting data can be fit directly
to a decaying exponential to extract the value of T2 . There is a slight complication to this. Since most of
the data points are not recorded, the time interval ∆t0 between adjacent points is no longer 1/SW. If ne is
the number of points acquired per echo, then ∆t0 = 2τ /ne . So when you analyze your data, (using Python,
for example) the array representing the time value of each data point should start at zero, and increase from
each point to the next by 2τ /ne .
Pulsed Nuclear Magnetic Resonance and Spin Echo
16
The effect of the local environment on the value of T1
The ability to detect the resonance requires some energy flow from the electronic circuit to the nuclear spins
and to the molecular environment. The strength of this “relaxation” process (characterized by the constant
T1 ) can be adjusted in water by adding various amounts of a paramagnetic salt, in this case copper sulfate.
Estimate the value of T1 of water containing the following concentrations of copper sulfate: zero (pure water),
2.5 mM, 5 mM, and 10 mM. (For pure water the relaxation time T1 is in the range of seconds).
To make these measurements, you will use a batch file called T1_IR_sweep.bat. A batch file is an editable
text file containing the values of the various parameters relevant to a given experiment. When one runs a
batch file (by double-clicking on it), it calls an executable file (in this case T1_IR_sweep.exe) with the
parameters specified in the batch file. The executable file runs the experiment and saves the results to one
or more data files.
What the program does is to first apply a π pulse, followed after time τ by a π/2 pulse and then record
the resulting FID. The program is repeated (NUMBER_OF_SCANS times) and the average of the FID’s is saved
to a file. The value of τ is then incremented by ∆τ , and this modified program is run, and the data saved to
a second file. This process is repeated until data have been saved for all specified values of τ . You will find
the batch files on the computer for the experiment (in a directory on the desktop), and you can find more
detailed explanations of the various parameters by reading that file.
When carrying out the T1 experiments, you should be sure that the time interval between repetitions of
the experiment is sufficiently large. If this time interval is not at least several times the value of T1 , your
results will be incorrect. The assumption in the analysis of this experiment is that before each repetition of
the experiment, the spins are in thermal equilibrium (that it, have reached their maximum polarization).
To analyze the data obtained with T1_IR_sweep you can use the program ispin_nmr_T1_analysis.py.
Instructions for using this program are given in comments at the beginning of the file. To keep track of
everything you have done collecting the T1 data, it is a good idea to keep a copy of the batch file, the data
files, and the python analysis file in the same directory.
Measure T1 of the four concentrations of copper sulfate (10 mM, 5 mM, 2.5 mM, and 1.25 mM), make
a plot of 1/T1 vs. concentration and interpret your results. Also measure the T1 of distilled water. Start
with the samples with higher concentrations of CuSO4 , since they have a smaller T1 , and therefore less
time is required to accumulate data. This way, any errors you make while figuring out how to make the
measurement will be less time consuming. Measuring the T1 is distilled water takes a long time, so I would
recommend not running this experiment during class time. The instructor will be glad to let you into the
lab and help you set up the experiment.
Observation of the “Stimulated Echo”
Another interesting experiment is the observation of the stimulated echo [3]. In a stimulated echo, one first
applies two rf pulses spaced time τ1 apart, followed by third rf pulse at time τ2 after the second pulse.
In addition to the usual spin-echos (how many would you expect there to be in this case), there will be a
stimulated echo at time τ1 after the third pulse. All three pulses are required for a stimulated echo to appear.
For some values of τ1 and τ2 , two stimulated echoes can be observed. A graphical method for predicting
when echos should occur is discussed in Appendix II.
To perform this experiment, you can use the batch file stimulated_echo.bat. You can set the time
intervals between the first two pulses pulses TAU_1 and between the second and third pulse TAU_2 and
the spectrometer records all the resulting echos (some of which will be “ordinary” spin echos, and some
stimulated echos).
To set up this experiment, it may be easier to first increase the inhomogeneity of the magnet to decrease
the duration of the FID and echo signals (you don’t want the duration of the Echo signals to be larger than
the time interval between pulses). To do this, you can change the position of the Z-gradient potentiometer
on the magnet controller so the duration of the FID is smaller than the time intervals between pulses (also
see the section on Measurement of Diffusion).
Make a measurement of the lifetime of the stimulated echo using the batch file Stimulated_echo_sweep.bat.
This file runs a program that does a sequence of experiments with different values of τ2 (and a fixed value
of τ1 ), and saves the resulting stimulated echo to a corresponding sequence of files.
Pulsed Nuclear Magnetic Resonance and Spin Echo
17
What is going on in the stimulated echo experiment to explain your lifetime measurements? You can try
looking in Slichter [4] for an explanation of the stimulated echo.
Measurement of diffusion
If one applies a magnetic field gradient to the sample, and individual spins move around due to diffusion,
one can use the spin-echo to measure the diffusion constant. You will make this measurement for a sample
of dionized water. The idea is the following: If the spins don’t move, but each spin experiences a different
magnetic field (due, for example, to a magnetic field gradient across the sample), then the spin-echo will
cause the spins at different locations, which get out of phase with each other after the first (π/2)-pulse to
rephase at time 2τ , when a π-pulse is applied at time τ . If the spins can move around, however, (due to
diffusion), then each spin will on average experience a different magnetic field after the π-pulse than before
the π-pulse. This will prevent the complete rephasing of the spins, and the echo amplitude will be reduced.
In this experiment you will apply a known magnetic field gradient to the sample of dionized water, and then
measure the echo lifetime using a sequence of spin echos (or Hahn echos), with different spacing τ between
the two pulses in the echo. By fitting the size of the echo as a function of pulse spacing τ , you will be able
to determine the (self) diffusion constant for water molecules.
In the paper of Carr and Purcell [5], the authors derive the expression for the echo signal as a function
of pulse delay in the 2-pulse spin echo sequence in the presence of a magnetic field gradient G. The size of
the echo at time t is
M (t) = M0 exp[(−t/T2 ) + (−γ 2 G2 Dt3 /12)],
(21)
where T2 is the spin-spin relaxation time, γ is the gyromagnetic ratio, G is the magnitude of the magnetic
field gradient, and D is the diffusion constant. For the effect of diffusion to dominate we require that for
times around t = T2 , the magnitude of the second term in the exponential of Eq. (21) dominates the first
term, or γ 2 G2 DT23 /12 ≥ T2 /T2 , or 12/γ 2 G2 D ≤ T23 . This means that we have to supply a sufficiently high
gradient field, so that the decay is mostly do to the gradient, rather than the effects of T2 . One can determine
if this condition is satisfied by making an independent measurement of T2 in the absence of a gradient field.
The details of the hardware indicate that a 0.1 Ω resistor is in series with the gradient coils, so the current
through the coils I is given by I = 10 (amp/volt) ∆V , where ∆V is the measured voltage difference between
the potentiometer position giving maximum field homogeneity and potentiometer position corresponding
to the inhomogeneous field. We also have (from the manufacturer’s specifications) that G ≡ ∂Bz /∂z =
6.6 [(µT/mm)/amp] I. This gives G = 66 [(µT/mm)/volt] ∆V .
Here are the steps:
• Apply a known gradient to the sample. This can be done by adjusting the Z-gradient potentiometer
on the magnet controller, and measuring the change in voltage ∆V across the 0.1 Ω resistor. The BNC
connector just below the potentiometer is connected across this resistor. Note that before doing this
procedure, you may first want to adjust the potentiometers to minimize the magnetic field gradients
across the sample, as described in Step 5 on page 12.
• Run the batch file Hahn_echo_sweep.bat, which produces a series of Hahn echos with varying value of
τ , the spacing between pulses. Let τ range from 10 ms to 150 ms in 10 ms steps. This can be done by
editing the batch file. Once you have collected the data, there is a python program available that you
can use to extract the echo sizes and fit the data, and from the fit, determine the diffusion constant.
• To analyze the data, first use the python analysis file “analyze_plot_ispin_echo_sweep_data.py”
to get the magnitude of the various echos as a function of delay τ . Then fit Eq. (21) to the data to
determine the value of D.
Prelab exersizes
1. What is meant by the “g-factor” for the proton?
2. What is the value of the proton g-factor?
Pulsed Nuclear Magnetic Resonance and Spin Echo
18
3. What is the energy difference between the spin states of the proton in a 1 Tesla magnetic field? What
frequency does this correspond to?
4. Derive Eq. (6) from Eq. (5).
5. For protons in a 1 T magnetic field, estimate the ratio n/N from Eq. (6). Is the approximation Eq. (7)
justified from this ratio?
6. Prove that Eq. (23) follows from Eq. (22) when As = A1 + iA2 .
7. (optional) Suppose you have a length ` of coaxial cable of characteristic impedance z0 and wave velocity
v0 . One end of the cable is connected to a load of impedance z at some frequency ω. What is the
impedance z 0 at frequency ω looking into the other end of the cable in terms of the quantities z0 , v0 ,
ω, `, and z. To solve this problem, express the current and voltage at any point in the cable as the
sum of two traveling waves: one traveling toward the end with impedance z, and the other traveling
away from the end with impedance z. Each of the two traveling waves has a well defined relationship
between the the current and voltage given by V = ±Iz0 , where the “+” applies to one traveling wave,
and the “−” applies to the other traveling wave (moving in the opposite direction). The sum of the
oppositely moving traveling waves gives the relationship between current and voltage, and hence, the
impedance at any point along the cable.
8. (optional) Use your result from Exersize 7 to obtain Eq. (18).
Appendix I: Analyzing signals as complex valued functions
General form of an arbitrary rf signal
The rf pulses generated by the spectrometer are at a frequency ω0 , which is relatively close to the resonant
frequency of the spins, ωs . That is, |ω0 − ωs | << ωs . These pulses are derived from a reference signal of
the form SR (t) = AR cos ω0 t = Re(AR eiω0 t ), where AR is the amplitude of the reference signal, Re(z) is the
real part of the complex number z, and Im(z) is the imaginary part of z. This reference signal is not only
used in the generation of the rf pulses, but also to process the signals generated in the pickup coil by the
precessing spins.
We assume the signal from the pickup-coil is in a narrow range of frequencies around the frequency ωs .
Any signal can be expressed as
S(t) = A1 (t) cos(ω0 t) + A2 (t) sin(ω0 t),
(22)
where A1 (t) and A2 (t) are slowly varying real-valued functions of t (that is, they vary much more slowly
than ω0 , ωs ). By defining As = A1 + iA2 , Eq. (22), can be more conveniently written
S(t) = Re[As (t)e−iω0 t ].
(23)
Note that all the information about the signal is contained in the real valued functions A1 (t) and A2 (t), or
more succinctly, in the complex valued function As (t). The two functions A1 (t) and A2 (t) will in general be 90
degrees out of phase with each other and oscillate at frequencies close to ωs −ω0 . To be more specific, suppose
that the spectrometer frequency ω0 /2π = 21.140 MHz and the spin precession frequency ωs /2π = 21.141
MHz. Then As (t) in Eq. (23) would have frequency components at (ωs − ω0 )/2π = 0.001 MHz = 1 kHz.
Choosing ω0 close to ωs significantly simplifies the task of storing the resulting signal. NMR signals in
liquids can last for many ms. If those signals are at a high frequency (say 10 MHz), then, according to the
Nyquist theorem, to preserve all the information in the signal one would have to digitize this signal at a rate
of at least twice this frequency, or at a rate of 20 MHz. If the signal lasts 100 ms, then 2 million data points
would have to be collected to store the signal. By mixing the signal down to 1 kHz, as in the above example,
only 2,000 data points would be needed to store the signal.
The functions A1 = Re(As ) and A2 = Im(As ) (with frequencies in the range of 1 kHz in the above
example) are generated by the spectrometer and saved to a file, which can then be read for displaying the
signal, or for further analysis. This appendix shows how the circuit in Figure 7 generates the signals A1 (t)
and A2 (t).
Pulsed Nuclear Magnetic Resonance and Spin Echo
19
Generating real and imaginary part of signal with mixers and phase shifters
Assume that we have a signal generated by the precessing spins (via the probe circuit), Sin (t) = Re[As (t)e−iω0 t ],
where As (t) is a slowly varying function of time (with frequency components close to ωs − ω0 ). Define
Ref[0◦ ] = cos ω0 t = Re e−iω0 t ,
Ref[90◦ ] = sin ω0 t = Re ie−iω0 t .
(24)
If we take the product Sin Ref[0◦ ] , and keep only the low frequency components, we find
Ref[0◦ ] × Sin (t)
=
low pass
−−−−−→
low pass
−−−−−→
(1/4)[e−iω0 t + eiω0 t ][As (t)e−iω0 t + A∗s (t)eiω0 t ]
(1/4)[As (t) + A∗s (t)]
(1/2)Re[As (t)]
(25)
We find also
Ref[90◦ ] × Sin (t)
=
low pass
−−−−−→
low pass
−−−−−→
(1/4)(ie−iω0 t − ieiω0 t )(As (t)e−iω0 t + A∗s (t)eiω0 t )
(1/4)(iA∗s (t) − iAs (t))
(1/2)Im[As (t)]
(26)
(27)
Appendix II: Diagrammatic approach to analyzing spin echos
Figure 12: Diagrammatic approach for analyzing
spin-echos [13]. The bottom of the graph shows the pulses
T
(vertical lines) and corresponding echos ( ). Below these symbols are the times when these events (pulses,
or echos) occur. In the upper part of the figure, horizontal lines represent protons in their low energy state
(spin pointing along the magnetic field), and the diagonal lines represent spins in their high energy state
(spin pointing opposite the magnetic field). Each rf pulse splits a line into two lines. An echo is generated
whenever two lines cross (denoted by black dots). Stimulated echos are those echos that wouldn’t exist
without all three pulses.
Pulsed Nuclear Magnetic Resonance and Spin Echo
20
Appendix III: Summary of different methods for obtaining data
from the iSpin NMR system
Using the LabView interface to communicate with the iSpin system
The developers of the iSpin system provide a graphical user interface (based on LabView) for specifying
pulses, and displaying and saving data, as shown in Figures 10 and 13. This interface allows the user to
Figure 13: LabView interface for communicating with the iSpin NMR system. Upper graph in the figure
shows the size of the FID signal as a function of the pulse duration. Lower curve shows the FID signal
corresponding to one of the points on the upper curve.
carry out several experiments. Examples are: Plotting and saving the FID from a single pulse (Fig. 10),
plotting signal size as a function of pulse duration (upper graph of Fig. 13), which is useful for determining the
pulse durations corresponding to 90◦ and 180◦ pulses, carrying out a CPMG (Carr-Purcell-Meiboom-Gill)
pulse sequence, and determining the exact resonance frequency of the spins.
Using batch files to perform an experiment using the iSpin system
A number of different experiments are available through the use of batch files. A batch file is an editable
text file containing the values of the various parameters relevant to a given experiment. When one runs a
batch file (by double-clicking on it), it calls an executable file with the parameters specified in the batch file.
The executable file runs the experiment and saves the results to one or more data files. The following is a
list of the various batch files, and what they do:
• singlepulse_nmr: Performs a single pulse experiment and saves the resulting FID to a file.
• T1_IR_sweep: T1 inversion-recovery– Applies a π pulse (to invert the spins), waits a varying time τ ,
applies a π/2 pulse and records the resulting FID. The size of the FID is a measure of how far the
spins have recovered toward equilibrium. The output of the program is a set of files each containing
Pulsed Nuclear Magnetic Resonance and Spin Echo
21
the recorded FID corresponding to a different value of τ . This experiment allows one to make a
determination of T1 .
• CPMG: This produces a set of pulses corresponding to a CPMG sequence, and records the resulting set
of echoes in a single file. This experiment can be used to determine T2 , and to measure the diffusion
constant.
• Hahn_echo: This produces a Hahn-echo sequence (π/2 − τ − π) that produces an echo centered around
time t = 2τ .
• Hahn_echo_sweep: This does a sequence of Hahn-echo experiments, saving the echo signals in a sequence of files, each corresponding to a given value of τ . This experiment can be used to determine
T2 .
• Stimulated_echo: This generates a sequence of three pulses with time interval τ1 between the first
two pulses, and time interval τ2 between the second two pulses, and records all the resulting echos.
Three of the resulting echos are Hahn echoes between the three pairs of pulses, but there are additional
echoes (one or two) that only occur in the presence of all three pulses.
• Stimulated_echo_sweep: This is similar to the stimulated echo, except that the time interval τ2 ,
between the second and third pulse is varied and the resulting stimulated echo is saved to a file for
each value of τ2 . This allows one to determine the lifetime of the stimulated echo.
References
[1] F. Bloch “Nuclear Induction”, Phys. Rev. 70, 460 (1946)
[2] A. Abragam, The principles of Nuclear Magnetism, Oxford (1961)
[3] E.L. Hahn, “Spin echoes” Phys. Rev. 80, 580, (1950)
[4] C.P. Slichter, Principles of Magnetic Resonance, Harper and Row (1963)
[5] H.Y. Carr and E.M. Purcell “Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance
Experiments”, Phys. Rev. 94, 630 (1954)
[6] Wikipedia, http://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion
[7] J.P. Hornak The Basics of NMR, http://www.cis.rit.edu/htbooks/nmr/ (1997 - 2011)
[8] E.R. Andrew, Nuclear Magnetic Resonance, Cambridge (1955)
[9] W. Low, Solid State Physics, Supplement 2, Sertz and Turnbull eds., Academic Press (1960)
[10] N. Ramsey, Molecular Beams, Oxford (1956)
[11] C. Cohen-Tannoudji and A. Kastler, Progress in Optics Vol 5, E. Wolf ed. (1966)
[12] A. Melissinos, Experiments in Modern Physics, Acedemic Press, 1966
[13] Lemberskiy and Xu, NMR lab report for Advanced Experimental Physics, Spring 2012.