# T2 Dependence on Time in Pulsed Nuclear Magnetic Resonance by adu47904

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```									             T2 Dependence on Time in Pulsed Nuclear Magnetic Resonance
Stephen Hurley
Physics Department, University of California, Santa Barbara

Abstract

The temperature dependencies of T2 as related to pulsed nuclear magnetic
resonance are measured and analyzed. Margarine is used as the sample and one plot of
data was taken from near boiling to near room temperature. There are various factors to
consider including the composition of the margarine. It is concluded that T2 may or may
not be affected by temperature.

Introduction

A brief history of pulsed nuclear magnetic resonance is in order. In 1946 nuclear
magnetic resonance was discovered by Edward Purcell and Felix Bloch. A few years
later, in 1950, Erwin Hahn accidentally discovered the idea of sending pulses to produce
an echo (see Theory).
Since then, pulsed nuclear magnetic resonance has been used in many commercial
and academic institutions for understanding the structure of materials. It is also
prominently used in MRI (magnetic resonance imaging) machines which are a less
invasive way of scanning the human body than traditional methods such as X-rays.
In the future, there will be many more applications of nuclear magnetic resonance
in the future.1

Theory

Pulsed nuclear magnetic resonance (PNMR) revolves around the spin of protons.
The idea is to change the spin via magnetic fields and observe the decay of the net spin in
a material. The x-y plane component of the average of the proton spins is observed on an
oscilloscope. This can give many details about the structure of the material.
The spin of the protons starts out in the +z direction. An effective magnetic field
is produced in the +x direction, making the spin rotate in the y-z plane at a rate of Ω. If
the magnetic field is turned on for only a moment, as in a pulse, the spin will turn specific
number of radians. The trick is to rotate the spin by π/2 or π so that the spin will end up
in the +y or -z directions, respectively.
In the π/2 position, the spins precess about the +z axis, rotating in the x-y plane.
Since there are a great number of protons, they do not all precess at the same rate. This
leads to phase dealignment, and as the spins spread out throughout the x-y plane, the net
spin becomes greatly diminished at a rate of

(1)

where T2 is called the Spin-Spin Relaxation Time. Unfortunately, it is hard to measure
this directly, because the magnetic field used to create the π/2 rotation is not uniform
itself. The magnetic field causes them to rotate at even greater differing rates, and so the
T2 recorded is not the real T2.1
The clever trick is to rotate again by π at a time τ so that the pulses that are
unaligned by the nonuniform magnetic field are rephased back. This creates an echo of
Mx,y at time 2τ that can be used to find T2. Then numbers can be plugged in and T2 can
be solved for giving

(2)

A diagram from the lab manual1 is shown here for better visualization:

Figure (1)

Method

The main apparatus is a magnet that forces the spins of the protons of a sample
that is placed in the middle (the middle is the most uniform of the magnetic field) to be
pointed in an upward direction (taken as the +z direction). This sets the sample up for
another magnetic field that will rotate the spins by the appropriate amounts.
The second apparatus is a voltage pulse generator. The generator can handle
pulses up to 15 MHz, which is sufficient for the purposes of the experiment. There are
settings available to set up sequences of pulses, such as setting up a arbitrary length pulse
of voltage followed by a independently arbitrary length pulse of voltage at a specific time
later. The pulse generator is connected to a coil that produces a magnetic field in the +x
direction.
The pulse generator is set up so that the first pulse of π/2 is sent (longer pulse
means greater radians). The second pulse equates to a π rotation. Then the interval
between pulses (τ) is adjusted so that the echo is at its highest height. The magnetic field
of the sample is converted into a voltage signal which is read on an oscilloscope. Then
using the ratio of the two voltage heights of Mx,y(0)=M0 and Mx,y(2τ), the equation is
used to solve for T2.
Before using a sample, a test sample was used to find the location (the “sweet
spot”) where the magnetic field is the greatest, since it is not uniform. This simply
involved moving the sample within the magnet and finding the highest height of the
The sample used in this experiment was margarine, because it was the only semi-
liquid substance within 10 feet of the experiment. It was first boiled using the only
heating device within 5 feet of the experiment, a soldering iron. Then it was immediately
placed in the magnet, and T2 was recorded multiple times along with the time at which
the T2 measurement was made with t=0 being just after the sample was placed in the
magnet.

Data

This is a plot of the time after boiling versus T2:
Time Vs. T2

160
140
120
100
T2 (ms)

Figure (2)                     80
60
40
20
0
0:00   2:24   4:48      7:12      9:36   12:00   14:24
Time (min)

The error bars are primarily from the inaccuracy of measuring the high points of
the voltages, since the shapes tended to be somewhat jagged. Other sources of error
include the inaccuracies in finding the sweet spot, because finding a maximum signal
involves trial and error method, which can easily lead to error. The last variable τ was
also selected to give the highest echo. Near the correct τ, the echo height plateaus, and so
the τ has some leeway in each direction giving a small amount of error as well.

Analysis

This experiment was primarily of a qualitative nature, since no actual
temperatures were recorded. The temperature, however, can be estimated from what is
known about temperature dependence over time. Here is a sketch of the temperature
overlaid with the data

Figure (3)
where the equation for the temperature is
(3)

This equation is roughly what the temperature graph might look like going from
boiling to room temperature. The T2 graph has a similar pattern after 1:12 minutes
suggesting a proportional dependency on temperature. Before 1:12, however, the graph
of T2 starts near 70 ms, suggesting a completely different dependency.

Discussion

When margarine is heated up to boiling, the temperature is not the only thing that
changes. At boiling and slightly below boiling, it is a very non viscous substance. As it
cools down, however, it becomes more viscous into the form that is most commonly
known as spreadable. T2 is known for being dependent on viscosity, and so this is a
major factor.3
One possible solution is that T2 is dependent on both temperature and viscosity. It
could be that after 1:12 the solution has reached an equilibrium viscosity, and the
temperature dependence takes prominence. More tests will be needed to conclude this
however.

Conclusion

The main conclusion is that more testing needs to be done. The margarine could
be dependent on temperature, viscosity, or both. It would be good to test a sample that
has viscosity independent of temperature, and therefore could be used to find a true
temperature dependence of T2.
If it is indeed true that T2 depends on temperature, this could affect many other
fields. For example, the industries that rely on T2 would need to make sure that the
temperature is stable throughout the readings. They would also need to make sure the
temperature does not go outside certain bounds which would too greatly affect the T2

References

1. Pulsed Nuclear Magnetic Resonance Lab Manual for Senior Lab at University of
California, Santa Barbara.

2. C. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963)

3. J. Seymour (jseymore@coe.montana.edu),
http://www.coe.montana.edu/che/jseymour/publicationsjds.htm, Montana State
University, 11-04.

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