Download Pulsed Nuclear Magnetic Resonance

Pulsed Nuclear Magnetic Resonance
Edited 4/5/17 by DGH, Stephen Albright & Prof. Mitrovic
Purpose:




Observe and understand pulsed nuclear magnetic resonance.
Measure relaxation processes in the time domain.
Study radio frequency (RF) resonant structures.
Become familiar with signal averaging techniques.
Pre-Lab Questions:
1. At a frequency ω, calculate the inductance L and impedance Z of a 7 turn coil with
a diameter of 0.8 cm and a length of 0.9 cm.
2. How could the setup in Figure 4 be used to find resonance in the Sample Coil
circuit? What would the oscilloscope signal look like at resonance? What would
the oscilloscope signal look like away from resonance?
3. What property of the magnetic field has to be determined in order to find the
diffusion coefficient?
I. Introduction
You will investigate pulsed nuclear magnetic resonance (PNMR). All nuclei with an odd
number of nucleons and some nuclei with an even number of nucleons have magnetic
moments. You will study the resonance of hydrogen-1, 1H. The gyromagnetic ratio of the
𝑘𝐻𝑧
proton is 𝛾 = 2𝜋 × 4.257 𝐺𝑎𝑢𝑠𝑠 = 2.675 × 108 rad/sT. (The magnetic moment of the
proton is about 660 times smaller than that of the electron.) Thus in a field of 1 T (10 kG)
the resonance frequency is 42.57 MHz.
The EPR experiment, which is also done in this course, has some similarities to the PNMR
experiment and some significant differences. In both experiments a spin resonance is
measured, however, since for a given magnetic field the gyromagnetic ratio of the proton
is very different from that of the electron, their resonant frequencies are very different. In
EPR the spins are bathed in a continuous (CW) microwave field while in a slowly ramped,
lightly modulated magnetic field. In this arrangement, a lock-in amplifier can be used to
observe resonance in the frequency domain. In PNMR the magnetic field is kept constant
but the field is pulsed. As a result, resonance occurs in the time domain and the transient
responses of all nuclear spins to the MW pulse will be observed. Commercial NMR
spectrometers, e.g. MRIs (magnetic resonance imaging), are more sophisticated versions
of this technique.
A look at the physics: Following is the equation of motion for a collection of magnetic
moments having a total magnetization M in a magnetic field:
𝑑𝑀
= −𝛾(𝑀 × 𝐻)
𝑑𝑡
or in component form with H parallel to the z-axis
𝑑𝑀𝑥
= −𝛾𝑀𝑦 𝐻𝑧
𝑑𝑡
(1𝑎)
𝑑𝑀𝑦
= 𝛾𝑀𝑥 𝐻𝑧
𝑑𝑡
(1𝑏)
𝑑𝑀𝑧
=0
𝑑𝑡
(1𝑐)
If the magnetization M of the sample is different from the thermal equilibrium magnetization M0 (which is parallel to the DC field), M will tend to relax back to M0 by various
processes in the material. Furthermore, if the magnetization in the x-y plane is not zero,
various interactions and processes will cause the x-y (transverse) magnetization to decay
to zero. We describe this mathematically by adding to Equation 1 relaxation terms to get
𝑑𝑀𝑥
𝑀𝑥
= −𝛾𝑀𝑦 𝐻𝑧 −
𝑑𝑡
𝑇2
(2𝑎)
𝑑𝑀𝑦
𝑀𝑦
= 𝛾𝑀𝑥 𝐻𝑧 −
𝑑𝑡
𝑇2
(2𝑏)
𝑑𝑀𝑧 𝑀0 − 𝑀𝑧
=
𝑑𝑡
𝑇1
(2𝑐)
These phenomenological relations are called the Bloch equations. T1 is the longitudinal
or spin-lattice relaxation time, and T2 is the transverse or spin-spin relaxation time. T1
measures how fast the magnetization returns to thermal equilibrium after being disturbed,
and T2 measures how rapidly the x-y magnetization decays. It can be shown that T2 ≤ T1.
Start at time t = 0 with the system in thermal equilibrium; Mz = M0 and Mx = My = 0.
Apply an RF pulse in the x-y plane at the Larmor precession frequency ω0 for a time τ’.
Assume the RF field is circularly polarized. In a coordinate system rotating at frequency
ω0, the field looks constant and directed along the y axis in the rotating coordinate
system. Then spins which in the DC field precess with frequency ω0 will now precess
about the y axis in the rotating coordinate system with a frequency
𝜔′ = 𝛾𝐻𝑅𝐹 .
(3)
2
𝜋
Suppose the pulse length τ’ is adjusted such that 𝜔′ 𝜏 ′ =
2
= 𝛾𝐻𝑅𝐹𝑥 𝜏′. A pulse of this
o
o
length is called a 90 pulse because the magnetization in the z direction is tipped by 90
relative to the 𝑥′ axis in the rotating coordinate system. How big do HRF and τ’ have to
−5
𝜋
be? If τ’ = 10 s then 𝐻𝑅𝐹 = [1/(2𝜋 × 4.257 × 103 × 10−5 )] = 6 G.
2
What happens after the pulse τ’ is removed? The magnetization in the laboratory frame is
now (taking x parallel to x’ at τ’ ) Mz =0, Mx = M0 and My = 0. Equation 2 tells us
𝑀𝑥 = 𝑀0 𝑒 −𝑡/𝑇2 cos 𝜔0 𝑡,
(4𝑎)
𝑀𝑦 = 𝑀0 𝑒 −𝑡/𝑇2 sin 𝜔0 𝑡,
(4𝑏)
𝑀𝑧 = 𝑀0 (1 − 𝑒
−𝑡⁄𝑇
1
).
(4𝑐)
With the appropriate electronic circuitry, we can detect this precessing magnetization Mx
o
and My. Note that if τ1 were increased to twice the value needed for a 90 pulse, then the
o
magnetization would have rotated by 180 and the state after removal of the pulse would
be Mx = My = 0 and Mz =  M0. The time dependence of Mz after the pulse would be
𝑀𝑧 = 𝑀0 (1 − 2𝑒
−𝑡⁄𝑇
1
).
II. The Resonant Circuit
To enhance the signal from the nuclear spins, the sample will be put inside the coil of the
resonant circuit arrangement shown in Figure 1. The effective impedance of the circuit is
given by 𝑍𝑇 = 𝑍𝐶2 || 𝑍𝑅 + 𝑍𝐶1 + 𝑍𝐿
Figure 1: Resonant circuit used to measure NMR. Note: C1 = 75pF & C2 = 560pF
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The capacitor C2 is the “coupling” capacitor for the tuned LC circuit formed by C1 and L.
The approximate resonant frequency of the resonant circuit is 𝜔0 ≈ 1/√𝐿𝐶1 . The
capacitor C2 matches the impedance of the tuned circuit to that of the transmission line
which allows optimal power coupling. The total impedance ZT which the circuit presents
to the line is found by:
𝑍𝑇 =
𝑍1 𝑍2
,
𝑍1 + 𝑍2
(5)
where at frequency ω
1
1
𝑍1 = 𝑅 − 𝑖 (
− 𝜔𝐿) , 𝑎𝑛𝑑 𝑍2 = −𝑖
.
𝜔𝐶1
𝜔𝐶2
(6)
Then
𝑍𝑇 =
1
𝑖
−𝑖 𝜔𝐶 (𝑅 − 𝜔𝐶 + 𝑖𝜔𝐿)
2
1
𝑖
𝑖
𝑅 − 𝜔𝐶 − 𝜔𝐶 + 𝑖𝜔𝐿
1
2
.
(7)
Using
1
1
1
= + .
𝐶𝑇 𝐶1 𝐶2
(8)
after some manipulation we find
𝑍𝑇 =
𝑅
1
1
− 𝜔𝐶 − 𝑖 [𝑅 2 + (𝜔𝐿 − 𝜔𝐶 ) (𝜔𝐿 − 𝜔𝐶 )]
2
1
𝑇
2
1
𝜔𝐶2 [𝑅 2 + (𝜔𝐿 − 𝜔𝐶 ) ]
𝑇
.
(9)
The real and imaginary components of the impedance, as well as the total impedance given
by Equation 9, are illustrated in Figure 2. At resonance the imaginary part of ZT vanishes
and impedance becomes real. For R ≪ 𝜔𝐿 and 𝐶1 ≪ 𝐶2 such that 𝐶𝑇 ~𝐶1, we can
approximate 𝜔 = 𝜔0 ≈ 1/√𝐶𝑇 𝐿, and
|𝑍𝑇 | ≈
1
.
𝑅(𝜔0 𝐶2 )2
(10)
To match the tuned circuit to the 50 Ω line we choose ZT = 50 Ω, so ω0C2 = 1/√50𝑅
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Figure 2: Impedance as a function of frequency for the circuit shown in Figure 1. The real
part of the impedance is indicated by the lightweight red solid line. The imaginary part of
the impedance is shown by the dashed line. The heavy blue solid line shows the total
impedance. Note: C1 = 75pF, C2 = 560pF, R  1Ω, & L  0.3µH
Note that when ω differs from ω0 by the amount
𝑅=(
1
2
1
∆𝜔 i.e. 𝜔 = 𝜔0 ± 2 ∆𝜔 such that
1
− 𝜔𝐿),
𝜔𝐶𝑇
(11)
then the total impedance is 1/2 of what it is at resonance, i.e.
|𝑍𝑇 | ≈
1
1
≈ 25𝛺
2 𝑅(𝜔0 𝐶2 )2
(12)
1
Eq. 11 with 𝜔 = 𝜔0 ± 2 ∆𝜔 can be written as
𝑅=
1
1
− (𝜔0 ± ∆𝜔 ) 𝐿
1
2
(𝜔0 ± 2 ∆𝜔 ) 𝐶𝑇
Substituting 𝜔0 = 1/√𝐶𝑇 𝐿, we get
±
∆𝜔
1
∆𝜔
(𝜔 𝐿) = 𝑅.
(
+ 𝜔0 𝐿) = ±
2𝜔0 𝜔0 𝐶𝑇
𝜔0 0
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Thus the quality factor is
𝑄=
𝜔0 𝐿 𝜔 0
=
.
𝑅
∆𝜔
(13)
The quantity Δω is called the full width at half maximum (FWHM).
To show what affect the tuned circuit makes to the current flowing through the coil consider
the following. If the coil were put directly across a voltage V (no capacitors), where 𝜔𝐿 ≫
𝑅, the current flowing through the inductor would be,
𝐼0 =
𝑉
𝑉
≈
.
𝑅 + 𝑖𝜔𝐿 𝜔𝐿
(14)
For the circuit in Figure 1 the current in the coil is
1
𝑉 [𝑅 + (𝜔𝐶 − 𝜔𝐿)]
𝑉
1
𝐼1 =
=
2 .
1
1
2
𝑅 − 𝑖 (𝜔𝐶 − 𝜔𝐿)
𝑅 + (𝜔𝐶 − 𝜔𝐿)
1
1
1
(15)
1
At resonance 𝜔𝐿 = 𝜔𝐶 ≈ 𝜔𝐶 and
𝑇
1
𝑉
𝐼1 ≈ .
𝑅
Thus I1 = I0(ωL/R) = QI0; the current in the resonant circuit is Q times larger than would
be flowing through the inductor alone with the same applied voltage. Note: Assuming the
power supply voltage is unaffected by the load, the impedance of the power supply and
capacitor C2 were completely neglected in this discussion of the current.
Using a similar argument, we can show that if a voltage were induced in the coil by the
precessing nuclear spins it surrounds, then the magnitude of that voltage is increased by
the Q of the tuned circuit.
III. Experimental Procedures
A) Observing the behavior of a resonant circuit:
Following are two ways to observe resonance in an RLC circuit. First, a manual and more
instructional approach. Construct the circuit shown in Figure 3, be sure the scope is set to
50Ω input impedance. Set the frequency generator to ~40MHz at -10dBm. Set the scope
to display a nice RF signal. In 0.1 MHz steps, adjust the frequency from ~39 – 41 MHz.
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Figure 3: A manual and instructive way to observe resonance
Over some narrow frequency range, the amplitude of the RF signal should drop abruptly.
Select smaller step sizes and scan until the frequency that gives the minimum amplitude
is identified to 3 or 4 decimal places; this is the resonant frequency of the circuit. Record
this frequency. Feel free to adjust the scope settings as needed.
Now an automated way to observe and find resonance. Construct the circuit shown in
Figure 4. Set the Sweep Time on the Wavetek Model 2001 Sweep Frequency Oscillator
(SFO) to Line. Use the horizontal dials near the top left and right of the SFO to set the
frequency range to ~25–55 MHz. Set the oscilloscope trigger to “AC Line” and the input
impedance to 1MΩ.
Figure 4: An automated way to observe and measure resonance
As its name implies, the output of the SFO sweeps a range of frequencies at a constant
output voltage. This sweeping range of radio frequencies (RF) is sent to the bridge and to
the resonant circuit. The signal reflected from resonant circuit is sent via the bridge through
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the detector where it is rectified. Note: the detector has a designated input and output. This
~DC signal is returned to the SFO along with the output from the signal generator. These
two signals are combined to create the Vertical Output of the SFO. On the scope this signal
will appear as a ~horizontal line representing the frequencies being swept and their
associated amplitudes. The lowest frequency is the left of the screen and the highest
frequency in on the right. There should be a pronounced and sharp dip in this line of swept
frequencies; this represents the resonant frequency. This reduction in amplitude is due to
the energy absorbed by the resonant circuit. Somewhere on this line of frequencies there
should be a single narrow sine pulse – this is the Marker. Its location shows the amplitude
at the frequency set on the frequency generator. Adjust the signal generator to place the
marker at the bottom of the dip. Record this value as the resonant frequency of the circuit.
1) Find the resonance width of the tuned circuit, and estimate its quality factor Q.
2) Given C1 = 75 pF, C2 = 560 pF and that the coil has 7 turns, a diameter of 0.8 cm
and a length of 0.9 cm. How does the coil inductance L measured compare with
that calculated in the pre-lab quiz?
o
−5
3) Compute the coil current necessary to rotate the proton spins by 90 in 10 s. What
is the voltage across the coil for such a current? What is the corresponding amplifier
output voltage?
B) Exploring the various circuit components:
1) Set the frequency synthesizer (Rhode & Schwartz) to -10 dBm (10 db below 1
milliwatt) and connect it directly to the oscilloscope. Change the input impedance
while observing the RF signal directly. Learn how to measure the frequency and
amplitude of the signal. Try different triggering options etc. See DPO3034
Digital Oscilloscope on the lab wiki under Equipment Manuals.
2) Explore the pulse generator. Assemble the circuit shown in Figure 5. Adjust and
observe the various outputs of the pulse generator. Construct both a single pulse
and double pulse sequence. Explore pulse widths from 4 to 30 µ sec and pulse
separations of ~1 ms. See DG535 Digital Delay & Pulse Generator on the lab
wiki under Equipment Manuals.
Figure 5: Circuit for exploring the pulse generator
3) Assemble Figure 6: Be sure to observe the input/output labels on the switch.
Change the scope’s input impedance while observing the signal on the scope.
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Figure 6: Circuit for exploring the switch.
4) Connect the output of the switch to the wide band amplifier and observe
amplifiers output while vary the scopes input impedance.
C) The PNMR circuit:
To understand this PNMR circuit, refer to Figure 7: The resonant circuit is designed to
resonate at a frequency near the frequency of the protons at 1T field.
1) Assemble the circuit shown in Figure 7. Set the frequency synthesizer to -10dBm
and to the resonant frequency found in part A).
Figure 7: The PNMR circuit
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2) Apply a 10-20 µsec wide pulse with a repetition rate of about 8 Hz to the switch.
To change the repetition rate on the pulse generator, click “Trig” twice. To get
back to the pulse length, simply press “Delay.”
3) The pulsed RF power from the wide band amplifier is fed through a noise
suppressor. This consists of a string of diode pairs connected in parallel with their
polarities reversed. The forward bias of the diodes suppresses noise from the
wide band amplifier during times when the switch is off. When the switch is on,
the amplified RF turns the diodes on and they offer very little signal loss.
4) TURN ON THE COOLING WATER FOR THE MAGNET. Turn on the magnet
and increase the current slowly. Set the time scale on the oscilloscope to 1 ms per
division. Vary the magnetic field until you observe the transient signal after the
pulse. The voltage in the sample circuit after the pulse is a decaying oscillation –
the oscillation is from the spins aligning with the coil and inducing a voltage. It
should be noted though that the detector output in Figure 7 is only the decay
envelope.
5) With the frequency synthesizer set to the resonant frequency found in part A),
adjust the DC magnetic field to maximize the PNMR decay. Be careful, there will
be several apparent maxima at different fields; be sure to find the absolute
maximum. Ask the TAs to show you a method of identifying the maximum using
the presence of beating in the signal. Note: for fine adjustment of the magnetic
field, use the small power adapter connected to the variable potentiometer.
6) Now vary the pulse width between 2 and 40 µs and measure the magnitude of the
PNMR signal. (WARNING: Make sure in this and subsequent measurements that
the RF amplifiers are not being saturated, i.e. the amplitude of the signal varies
linearly with gain. The maximum amplitude of the PNMR signal as measured on
the oscilloscope should be no larger than 1 volt). Take a sufficient number of points
to be able make a good plot of signal size versus pulse width. Comment briefly.
7) When the transient signal is at a maximum with the lowest value of pulse width,
the product of the length of the pulse, tp, and the strength of the RF field HRF
produces a 90° rotation of the spin magnetization, i.e. a 90° pulse given by
𝜋
= 𝜔𝑡𝑝 = 𝛾𝐻𝑅𝐹 𝑡𝑝
2
8) Vary the pulse width and determine how the power from the frequency
synthesizer must be changed to maintain a 90° pulse. (WARNING: Make sure
you do not confuse the maxima associated with a 270° pulse with those of a 90°
pulse). Plot pulse width versus power to maintain a 90° pulse and comment.
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D) Magnetic Field Gradient:
The magnetic field between the magnet’s pole faces is only homogeneous enough to
optimally support NMR over the central ~1” diameter of the pole faces. Move the NMR
shielding box vertically along the graduated vertical post and observe the effects of a field
gradient on the decay of the transient signal. You will note that as the field gradient
increases, the rate at which the transient signal decays increases. This occurs because the
nuclear spins in different parts of the sample are in slightly different DC magnetic fields,
which causes their Larmor precession frequencies to be slightly different. With a larger
difference in precession frequencies, the magnetization in the x-y plane dephases more
rapidly. The field inhomogeneity is essentially influencing the apparent T2 in Equations 2
and 4.
To observe the effect of a field gradient on the decay envelope, perform the following
experiment:
1) Setup the maximum 90° pulse found in section C) & record the decay time.
2) Raise the shielding box 1mm – as measure against the graduated vertical post - &
record the decay time.
3) Continue this procedure for several heights using 1 or 2 mm steps. Max height is
6 cm. Use this data to determine the field gradient in units of T/mm.
4) Plot decay time as a function of position.
When the field gradient is large and the transient decay is very short, you will note that the
PNMR signal appears only about 30 µs after the pulse is turned on. This is because the
large RF signal of the pulse saturates the sensitive amplifiers, which take some time to
recover.
The oscilloscope you are using has a variety of convenient features that can simplify and
enhance the quality of your measurements. These include cursors, signal averaging and
several other useful features - consult the scope manual for instruction. If after reading the
manual you still need help, consult the TA.
E) Measurement of T1:
o
The transverse component of the magnetization immediately after a 90 pulse is equal in
magnitude to the z component of the magnetization just before the pulse is applied. Also,
immediately after the pulse, the z component is zero and recovers with time according to
Equation 4 as
𝑀𝑧 = 𝑀0 (1 − 𝑒
−𝑡⁄𝑇
1
).
o
Now suppose we apply another 90 pulse after the first but before the z component has had
an opportunity to fully recover to the equilibrium value M0. If the time between pulses is
tA, the magnetization along z at the time of the second pulse is
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𝑀𝑧 = 𝑀0 (1 − 𝑒
𝑡
− 𝐴⁄𝑇
1 ).
After this pulse the transverse magnetization is then MzA which is different than M0.
Therefore, if we measure the transient signal as a function of time between pulses, we can
determine the relaxation time T1.
1) Setup the maximum 90° pulse found in section C). Use a BNC-T adapter on A-B
out and add a second 90° pulse from the C-D out.
2) Set the delay between the two pulses to ~10ms and record the amplitude of the
transient signal after the second pulse. Systematically decrease the delay between
the pulses and record the corresponding amplitude of the second transient signal.
3) Plot the results to obtain the spin-lattice relaxation time, T1. When the delay is
short, it may be desirable to apply a small field gradient to keep the decay of the
first pulse from overlapping with the decay of the second pulse.
Figure 8: Two 90° Pulses. Note: The small bump to the right of the
display indicates the pulse width determined in step C) 6 is not
accurate. To remove this bump, slightly adjust the pulse width.
F) Spin Echoes:
We now proceed to a more complex series of pulses. We first apply a 90° pulse to the
spins; then after waiting a time τ we apply a second pulse twice as wide as the first
(180°). This second pulse will rotate the spins an additional 180°. What happens?
The 90° pulse rotates the magnetization from the z axis to the xʹ axis in a frame rotating
with the Larmor precession frequency (and the frequency of the applied RF field) as
illustrated in Figure 9. After the pulse is over, the spins dephase with respect to one another
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(due to inhomogeneity in the magnetic field) and in so doing the transient signal disappears
(Figure 9.c.). Now after a time τ we apply a second pulse, this one 180°. This pulse rotates
all the spins by 180° about the yʹ axis, i.e. it flips over the pancake of dephased spins in the
x-y plane. This is shown in Figure 9.d. Remember, T2 ≤ T1.
Now, those spins that were leading in phase are now trailing, while those spins that were
behind before the second pulse are now leading. Those spins now behind are still
precessing faster however, while those now ahead are precessing slower. Therefore, they
all come back into phase at a time τ after the second pulse (Figure 9.e.). The transverse
magnetization is recovered and a signal is detected, called the spin echo.
The discussion above provides a physical picture of how it is that an echo forms after a
pair of pulses, but it does not explain the possible influence of various parameters on the
amplitude of the echo. Firstly, in this discussion we have neglected the effects of
relaxation. Insofar as there are relaxation or dephasing processes (T2) which do not depend
on a static magnetic field gradient then the amplitude of the echo will be diminished
−𝑡
proportional to 𝑒 ⁄𝑇2 where 𝑡 = 2𝜏. Secondly, we have assumed the spins remain
stationary in the magnetic field such that their precession frequency is a constant in time.
But we are observing protons in water, which are free to diffuse. Since a particular spin
may therefore precess at somewhat different frequencies before and after the second pulse
as a result its diffusive motion, the spins will not all come back into phase at the same time.
It can be shown that for a sample having a diffusion coefficient D in a field gradient G the
amplitude of the spin echo for a 90°-180° pulse sequence is
𝑀𝑒𝑐ℎ𝑜 (𝑡 = 2𝜏) = 𝑀0 𝑒
−2𝜏⁄𝑇
2
𝑒 −(
2⁄ )𝑘𝜏3
3
,
where
𝑘 = 𝛾 2 𝐺 2 𝐷.
d)
e)
Figure 9. Time evolution of PNMR spins: a) Before first pulse b) After first pulse
c) Before second pulse, with some dephasing in the x-y plane d) After second pulse,
with all spins flipped e) Spin echo, after spins have realigned.
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Observe the amplitude of the spin echo following a 90°-180° pulse sequence. It is
important that the resonant frequency, ωo, be set correctly. Vary the frequency of the
synthesizer slightly and set it at the value that maximizes the echo.
1) Measure the echo amplitude as a function of time between the first and
second pulses for a specific value of the magnetic field gradient (for
example, a height of 2mm). Measure at ~ ten different delay times. Increase
the delay time until the echo’s magnitude is reduced by a factor of 30 as
compared to the magnitude at shorter times. Note: if 𝜏 is too small, the
decay of the second pulse may become unstable. Do not record unstable
data. If needed, the signal averaging function on the scope will help clean
up noisy signals.
2) Repeat step 2) for a second value of field gradient (say a height of 4mm).
Consider how to plot the data in order to determine T2, D and M0. (You
might consider log (Mecho /M0)/τ versus 𝜏 2 for several different choices of
M0, but there are other procedures that will work as well).
3) Select a particular value of 𝜏 such that the echo amplitude decreases by an
order of magnitude or more when the maximum field gradient is applied.
Measure the echo amplitude at different field gradients. Again use signal
averaging before recording data. Make a plot of the appropriate functions of
echo amplitude and field gradient to obtain a linear graph. From this,
determine the diffusion coefficient D.
4) Compare your measured values of D to the accepted value, D = 2.5 × 10-5
cm2/5 at 25°C.
References:
Note that Hahn, Free Nuclear Induction is located in the references folder on the lab
wiki.
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