Download Pulsed Nuclear Magnetic Resonance to Obtain Characteristic Times

Pulsed Nuclear Magnetic Resonance to Obtain Characteristic Times for
Mineral Oil and Gycerin
Callie Fiedler,a) David Mallin,b) and Andy Vescib)
(Dated: 27 March 2011)
Using Pulsed Nuclear Mag thenetic spectroscopy the Spin-Lattice (T1 ) and Spin-Spin (T2 ) relaxation times
of Glycerin and Mineral Oil. The T1 for Glycerin and Mineral Oil were 40.4 ± 0.1 and 20.2 ± 0.1 respectively,
and T2 for Glycerin and Mineral Oil were 35.7 ± 0.1 and 15.8 ± 0.1 respectively. The Free Induction Decay
(T∗2 ) was 1.2 ± 0.1 for Glycerin and 1.1 ± 0.1 for Mineral Oil.
I.
INTRODUCTION
Nuclear Magnetic Resonance (NMR) was observed by
Edward Purcell and Felix Bloch, independently, in 1946
when investigating the behavior of nuclei once placed in
an external, uniform magnetic field and subject to a radio
frequency (RF) magnetic field oriented perpendicular to
the external field 1 .
While there have been many advancements made in
the sciences due to the advancements afforded by the discovery of NMR, the most notable contribution to society
resides in the medical field. NMR or, in medical terms,
Magnetic Resonance Imaging (MRI) is the most current
and effective method used in medicine to non-invasively
investigate what is physically happening within living organisms. In order to observe the inner workings of living
systems one must understand the characteristic times associated with the internal molecules and their exposure
to an external magnetic field.
This experiment will investigate two characteristic
times, relaxation times, associated with mineral oil and
glycerin, two molecules with very prevalent structures
to the function of not only the human body, but also
our understanding of living structures in general. More
specifically, the characteristic relaxation time related to
an atoms interaction with surrounding molecules (T1 )
and the relaxation time associated with the internal interactions of the atom (T2 ) are determined.
This paper will first address the theoretical concepts
essential to this experiment(II) and discuss the physical apparatus we used to conduct research (III). I then
discuss the design of our experiments (IV), first the SpinLattice Relaxation Time (T1 ) for both Glycerin and Mineral Oil (A) and the the Spin-Spin Relaxation Time (T2 )
for Glycerin and Mineral Oil (B) and the Free Induction Decay (T∗2 ) of both. The paper terminates with a
discussion of the results found (V).
a) Also
at Physics Department, University of San Diego.
Department, University of San Diego.
b) Physics
II.
THEORY
A.
Nuclear Resonance
To obtain Nuclear Magnetic Resonance, the nuclei in
investigation must have a nonzero magnetic moment (µ)
and angular momentum (J) related by
µ = γJ,
(1)
where γ is the gyromagnetic ratio, which equals 2.657
×104 radians per second-gauss for a proton. The angular
momentum is quantized in terms of ~ and I, nuclear spin,
J = ~I.
(2)
When exposed to an external magnetic field there is a
certain energy associated to the nucleus. In particular,
when a proton resides in an external magnetic field in the
z-direction, the magnet energy is
U = −µz B0 = −γ~Iz B0 ,
(3)
where Iz is the nuclear spin in the z-direction and B0 is
the external magnetic field. For a proton, I = 1/2 and so
it follows that the allowed spins in the z-direction are mI
= ±1/2 because quantum mechanics requires that these
values be quantized such that mI = I, I−1, I−2, · · · −I.
When nuclei with these characteristics are subjected to
a uniform external magnetic field, the allowed ground
energy states split in two. The energy difference between
these two states, ∆U is proportional angular frequency,
∆U = ~ω0 = γ~B0 ,
(4)
where ω0 = γB0 is the fundamental resonance condition.
When a proton is exposed to a constant magnetic field
and a perpendicular rf magnetic field at its resonant frequency, the energy created is sufficient to overcome ∆U
and spin transitions between mI = ±1/2 can be driven,
i.e. the nuclear spin of the atoms can be flipped from
”up” to ”down” or vice versa.
Pulsed Nuclear Magnetic Resonance (PNMR) allows
for one to determine characteristic relaxation times of
certain molecular structures, e.g. mineral oil and glycerin. PNMR utilizes the same techniques of classic NMR
with the addition of rf pulses with specific widths (90o or
180o ) that alter the next magnetization of the molecule 2 .
2
These pulses tip the nuclear spins into to x-y plane that
then precess around the B0 magnetic field, which create a time varying voltage in a x-y plane magnetization
pick-up coil.
and
dMy
My
=−
,
dt
T2
(7)
whose solutions are
B.
Spin-Lattice Relaxation
t
My (t) = Mx (t) = M0 e− dt ,
One of the relaxation times in interest is the spinlattice relaxation time (T1 ). When an isolated proton
has its spin flipped, this magnetization can be indefinitely
maintained. Because molecules have many protons which
have varying magnetizations in the plane (all averaged to
have a net flipped spin), the spins realign to the external,
primary magnetic field (z-direction oriented). The time
it takes for the spins in the molecule to realign to thermal
equilibrium magnetization is defined as T1 .
To determine T1 the net magnetization (M0 ) of the
molecule is tipped 180o to -M0 . To ensure one is applying a true 180o pulse, the free induction decay (FID)
should be minimized, corresponding to a minimized voltage amplitude within the molecule in question. A pick-up
coil monitors the return of the net magnetization back to
M0 , thermal equilibrium magnetization. A second pulse
of 90o is introduced to determine if the net magnetization
in the x-y plane has gone to zero, indicating the return to
thermal equilibrium. Contrary to the 180o pulse, to ensure one is applying a true 90o pulse the FID amplitude
must be maximized. By interrogating the net spin at
varying increments using the pick-up coil, researchers can
determine how greatly the magnetization has changed in
the molecule at varying delay times. By plotting the
maximized FID amplitudes, M, versus the delay times,
one can observe the relaxation through the dying exponential
M (t) − M0
dMz
=
,
dt
T1
(8)
where t represents twice the delay time and T2 is the
spin-spin relaxation time.
III.
A.
EXPERIMENTAL DESIGN
Apparatus
For our experiment we utilized an oscilloscope, an RF
source, and a permanent magnet system to implement
and observe the effects of PNMR on different molecular
samples. While the oscilloscope provides the necessary
electronic feedback to observe the effects of the PNMR,
the intricacies of the magnet configuration create the desired effect. As seen in Figure III A, the permanent magnet is the most external component to the magnet system, which creates a net magnetization of the molecular
structure of the sample in the z-direction, e.g. mineral
oil and glycerin 3 .. An rf coil configuration is positioned
around the sample to allow for the necessary pulses into
the x-y plane, creating a magnetic field in the x-direction.
To observe the effects of PNMR, a pick-up coil is
wound directly around the sample cell to create a magnetic field perpendicular to both the external, permanent
magnet and the rf magnetic source in the y-direction.
(5)
where dMz /dt indicates the net change in magnetization
in the z-direction.
C.
Spin-Spin Relaxation
The spin-spin relaxation time (T2 ) is the time it takes
for the spins to be altered from their non-thermal equilibrium magnetization in the x-y plane. Unlike the spinlattice relaxation time, to measure T2 one must first use
a 90o pulse, when the FID amplitude is maximized. Because the interactions of the individual spins within an
atom vary their local magnetic field and the frequency
at which the spins precess back to a non-thermal equilibrium from being pulsed, the spins dephase. The dephasing of spins creates a variance in the voltage being
interpreted by the rf pick-up coil. The rate at which the
net magnetization in the x-y plane is represented by
dMx
Mx
=−
dt
T2
(6)
FIG. 1. A top-view depiction of the magnetic system used
to conduct our research. The center region is the location in
which the sample is placed.
3
IV.
RESULTS
T2 can be extracted from the exponential and was 15.8
± 0.1.
The T1 for Glycerin 40.4 ± 0.1 from our plot of the
voltage detected from the pick-up coil versus the different
variances of the delay time, as seen in Figure ??. The
Mineral Oil spin-lattice relaxation time value was 20.2 ±
0.1 from the exponential depicted in Figure IV.
FIG. 4. Graph of the spin-spin relaxation time voltage values for glycerin versus the changing delat time, displaying a
decaying exponential.
FIG. 2. Graph of the spin-lattice relaxation time voltage values versus the changing delay time, displaying an inverse decaying exponential, from which you can obtain the T1 value
for Glycerin.
FIG. 5. Graph of the spin-spin relaxation time voltage values
for mineral oil versus the changing delat time, displaying a
decaying exponential.
V.
DISCUSSION
Our results were significantly similar to those of previous literature 1 . For both mineral oil and glycerin had T1
values lower than the observed T2 values, as expected.
FIG. 3. Graph of the spin-lattice relaxation time voltage values for Mineral Oil versus the changing delat time, displaying
a decaying exponential.
VI.
REFERENCES
1 Klein,
The spin-spin relaxation time were in accordance with
the literature for glycerin in that we obtained a T2 of
35.7 ± 0.1. Similar to the method used to determine the
T1 value, this value can be expressed through the dying
exponential as seen in Figure ??. As seen in ??, the
W. Nuclear Magnetic Resonance: Free- Induction Decay
and Spin Echoes in a 0.05-T Magnetic Field. American Journal
of Physics. 1989
2 Melissinos, Adrian C. and Jim Napolitano. Mag- netic Resonance
Experiments. . Experiments in Modern Physics. 2008
3 Wolff-Reichert, Barbara. A Conceptual Tour of TeachSpins
Pulsed NMR. 2003.