Chapter 3 SPIN PHYSICS
Spin Properties of Spin Nuclei with Spin Energy Levels Transitions Energy Level Diagrams 
Boltzmann Statistics Spin Packets T_{1} Processes Precession T_{2} Processes Rotating Frame of Reference Pulsed Magnetic Fields Spin Relaxation Spin Exchange Bloch Equations
Spin
What is spin? Spin is a fundamental property of nature like electrical charge or mass. Spin comes in multiples of 1/2 and can be + or . Protons, electrons, and neutrons possess spin. Individual unpaired electrons, protons, and neutrons each possesses a spin of 1/2.
In the deuterium atom ( ^{2}H ), with one unpaired electron, one unpaired proton, and one unpaired neutron, the total electronic spin = 1/2 and the total nuclear spin = 1.
Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin. An example is helium. In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance.
Properties of Spin
When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon, of frequency . The frequency depends on the gyromagnetic ratio, of the particle.
= B
For hydrogen, = 42.58 MHz / T.
Nuclei with Spin
The shell model for the nucleus tells us that nucleons, just like electrons, fill orbitals. When the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, and 126, orbitals are filled. Because nucleons have spin, just like electrons do, their spin can pair up when the orbitals are being filled and cancel out. Almost every element in the periodic table has an isotope with a non zero nuclear spin. NMR can only be performed on isotopes whose natural abundance is high enough to be detected. Some of the nuclei routinely used in NMR are listed below.
Nuclei

Unpaired Protons

Unpaired Neutrons

Net Spin

(MHz/T)

^{1}H

1

0

1/2

42.58

^{2}H

1

1

1

6.54

^{31}P

1

0

1/2

17.25

^{23}Na

1

2

3/2

11.27

^{14}N

1

1

1

3.08

^{13}C

0

1

1/2

10.71

^{19}F

1

0

1/2

40.08
 Energy Levels
To understand how particles with spin behave in a magnetic field, consider a proton. This proton has the property called spin. Think of the spin of this proton as a magnetic moment vector, causing the proton to behave like a tiny magnet with a north and south pole.
When the proton is placed in an external magnetic field, the spin vector of the particle aligns itself with the external field, just like a magnet would. There is a low energy configuration or state where the poles are aligned NSNS and a high energy state NNSS.
Transitions
This particle can undergo a transition between the two energy states by the absorption of a photon. A particle in the lower energy state absorbs a photon and ends up in the upper energy state. The energy of this photon must exactly match the energy difference between the two states. The energy, E, of a photon is related to its frequency, , by Planck's constant (h = 6.626x10^{34} J s).
E = h
In NMR and MRI, the quantity is called the resonance frequency and the Larmor frequency.
Energy Level Diagrams
The energy of the two spin states can be represented by an energy level diagram. We have seen that = B and E = h , therefore the energy of the photon needed to cause a transition between the two spin states is
E = h B
When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs.
In the NMR experiment, the frequency of the photon is in the radio frequency (RF) range. In NMR spectroscopy, is between 60 and 800 MHz for hydrogen nuclei. In clinical MRI, is typically between 15 and 80 MHz for hydrogen imaging.
CW NMR Experiment
The simplest NMR experiment is the continuous wave (CW) experiment. There are two ways of performing this experiment. In the first, a constant frequency, which is continuously on, probes the energy levels while the magnetic field is varied. The energy of this frequency is represented by the blue line in the energy level diagram.
The CW experiment can also be performed with a constant magnetic field and a frequency which is varied. The magnitude of the constant magnetic field is represented by the position of the vertical blue line in the energy level diagram.
Boltzmann Statistics
When a group of spins is placed in a magnetic field, each spin aligns in one of the two possible orientations.
At room temperature, the number of spins in the lower energy level, N^{+}, slightly outnumbers the number in the upper level, N^{}. Boltzmann statistics tells us that
N^{}/N^{+} = e^{E/kT}.
E is the energy difference between the spin states; k is Boltzmann's constant, 1.3805x10^{23} J/Kelvin; and T is the temperature in Kelvin.
As the temperature decreases, so does the ratio N^{} /N^{+}. As the temperature increases, the ratio approaches one.
The signal in NMR spectroscopy results from the difference between the energy absorbed by the spins which make a transition from the lower energy state to the higher energy state, and the energy emitted by the spins which simultaneously make a transition from the higher energy state to the lower energy state. The signal is thus proportional to the population difference between the states. NMR is a rather sensitive spectroscopy since it is capable of detecting these very small population differences. It is the resonance, or exchange of energy at a specific frequency between the spins and the spectrometer, which gives NMR its sensitivity.
Spin Packets
It is cumbersome to describe NMR on a microscopic scale. A macroscopic picture is more convenient. The first step in developing the macroscopic picture is to define the spin packet. A spin packet is a group of spins experiencing the same magnetic field strength. In this example, the spins within each grid section represent a spin packet.
At any instant in time, the magnetic field due to the spins in each spin packet can be represented by a magnetization vector.
The size of each vector is proportional to (N^{+}  N^{}).
The vector sum of the magnetization vectors from all of the spin packets is the net magnetization. In order to describe pulsed NMR is necessary from here on to talk in terms of the net magnetization.
Adapting the conventional NMR coordinate system, the external magnetic field and the net magnetization vector at equilibrium are both along the Z axis.
T_{1} Processes
At equilibrium, the net magnetization vector lies along the direction of the applied magnetic field B_{o} and is called the equilibrium magnetization M_{o}. In this configuration, the Z component of magnetization M_{Z} equals M_{o}. M_{Z} is referred to as the longitudinal magnetization. There is no transverse (M_{X} or M_{Y}) magnetization here.
It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make M_{Z}=0.
The time constant which describes how M_{Z} returns to its equilibrium value is called the spin lattice relaxation time (T_{1}). The equation governing this behavior as a function of the time t after its displacement is:
M_{z} = M_{o} ( 1  e^{t/T}^{1} )
T_{1} is therefore defined as the time required to change the Z component of magnetization by a factor of e.
If the net magnetization is placed along the Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T_{1}. The equation governing this behavior as a function of the time t after its displacement is:
M_{z} = M_{o} ( 1  2e^{t/T}^{1} )
The spinlattice relaxation time (T_{1}) is the time to reduce the difference between the longitudinal magnetization (M_{Z}) and its equilibrium value by a factor of e.
Precession
If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency.
T_{2} Processes
In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets.
The time constant which describes the return to equilibrium of the transverse magnetization, M_{XY}, is called the spinspin relaxation time, T_{2}.
M_{XY} =M_{XYo} e^{t/T}^{2}
T_{2} is always less than or equal to T_{1}. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have M_{o} along Z.
Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T_{1} governs the rate of recovery of the longitudinal magnetization.
In summary, the spinspin relaxation time, T_{2}, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T_{2} and T_{1} processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T_{2} is less than or equal to T_{1}.
Two factors contribute to the decay of transverse magnetization.
1) molecular interactions (said to lead to a pure pure T_{2} molecular effect)
2) variations in B_{o} (said to lead to an inhomogeneous T_{2} effect
The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T_{2} star and is given the symbol T_{2}*. The relationship between the T_{2} from molecular processes and that from inhomogeneities in the magnetic field is as follows.
1/T_{2}* = 1/T_{2} + 1/T_{2inhomo}.
Rotating Frame of Reference
We have just looked at the behavior of spins in the laboratory frame of reference. It is convenient to define a rotating frame of reference which rotates about the Z axis at the Larmor frequency. We distinguish this rotating coordinate system from the laboratory system by primes on the X and Y axes, X'Y'.
A magnetization vector rotating at the Larmor frequency in the laboratory frame appears stationary in a frame of reference rotating about the Z axis. In the rotating frame, relaxation of MZ magnetization to its equilibrium value looks the same as it did in the laboratory frame.
A transverse magnetization vector rotating about the Z axis at the same velocity as the rotating frame will appear stationary in the rotating frame. A magnetization vector traveling faster than the rotating frame rotates clockwise about the Z axis. A magnetization vector traveling slower than the rotating frame rotates counterclockwise about the Z axis .
In a sample there are spin packets traveling faster and slower than the rotating frame. As a consequence, when the mean frequency of the sample is equal to the rotating frame, the dephasing of M_{X'Y'} looks like this.
Pulsed Magnetic Fields
A coil of wire placed around the X axis will provide a magnetic field along the X axis when a direct current is passed through the coil. An alternating current will produce a magnetic field which alternates in direction.
In a frame of reference rotating about the Z axis at a frequency equal to that of the alternating current, the magnetic field along the X' axis will be constant, just as in the direct current case in the laboratory frame.
This is the same as moving the coil about the rotating frame coordinate system at the Larmor Frequency. In magnetic resonance, the magnetic field created by the coil passing an alternating current at the Larmor frequency is called the B_{1} magnetic field. When the alternating current through the coil is turned on and off, it creates a pulsed B_{1} magnetic field along the X' axis.
The spins respond to this pulse in such a way as to cause the net magnetization vector to rotate about the direction of the applied B_{1} field. The rotation angle depends on the length of time the field is on, , and its magnitude B_{1}.
= 2 B_{1}.
In our examples, will be assumed to be much smaller than T_{1} and T_{2}.
A 90^{o} pulse is one which rotates the magnetization vector clockwise by 90 degrees about the X' axis. A 90^{o} pulse rotates the equilibrium magnetization down to the Y' axis. In the laboratory frame the equilibrium magnetization spirals down around the Z axis to the XY plane. You can see why the rotating frame of reference is helpful in describing the behavior of magnetization in response to a pulsed magnetic field.
A 180^{o} pulse will rotate the magnetization vector by 180 degrees. A 180^{o} pulse rotates the equilibrium magnetization down to along the Z axis.
The net magnetization at any orientation will behave according to the rotation equation. For example, a net magnetization vector along the Y' axis will end up along the Y' axis when acted upon by a 180^{o} pulse of B_{1} along the X' axis.
A net magnetization vector between X' and Y' will end up between X' and Y' after the application of a 180^{o} pulse of B_{1} applied along the X' axis.
A rotation matrix (described as a coordinate transformation in #2.6 Chapter 2) can also be used to predict the result of a rotation. Here is the rotation angle about the X' axis, [X', Y', Z] is the initial location of the vector, and [X", Y", Z"] the location of the vector after the rotation.
Spin Relaxation
Motions in solution which result in time varying magnetic fields cause spin relaxation.
Time varying fields at the Larmor frequency cause transitions between the spin states and hence a change in M_{Z}. This screen depicts the field at the green hydrogen on the water molecule as it rotates about the external field B_{o} and a magnetic field from the blue hydrogen. Note that the field experienced at the green hydrogen is sinusoidal.
There is a distribution of rotation frequencies in a sample of molecules. Only frequencies at the Larmor frequency affect T_{1}. Since the Larmor frequency is proportional to B_{o}, T_{1} will therefore vary as a function of magnetic field strength. In general, T_{1} is inversely proportional to the density of molecular motions at the Larmor frequency.
The rotation frequency distribution depends on the temperature and viscosity of the solution. Therefore T_{1} will vary as a function of temperature. At the Larmor frequency indicated by _{o}, T_{1} (280 K ) < T_{1} (340 K). The temperature of the human body does not vary by enough to cause a significant influence on T_{1}. The viscosity does however vary significantly from tissue to tissue and influences T_{1} as is seen in the following molecular motion plot.
Fluctuating fields which perturb the energy levels of the spin states cause the transverse magnetization to dephase. This can be seen by examining the plot of B_{o} experienced by the red hydrogens on the following water molecule. The number of molecular motions less than and equal to the Larmor frequency is inversely proportional to T_{2}.
In general, relaxation times get longer as B_{o} increases because there are fewer relaxationcausing frequency components present in the random motions of the molecules.
Spin Exchange
Spin exchange is the exchange of spin state between two spins. For example, if we have two spins, A and B, and A is spin up and B is spin down, spin exchange between A and B can be represented with the following equation.
A() + B() A() + B()
The bidirectional arrow indicates that the exchange reaction is reversible.
The energy difference between the upper and lower energy states of A and of B must be the same for spin exchange to occur. On a microscopic scale, the spin in the upper energy state (B) is emitting a photon which is being absorbed by the spin in the lower energy state (A). Therefore, B ends up in the lower energy state and A in the upper state.
Spin exchange will not affect T_{1} but will affect T_{2}. T_{1} is not effected because the distribution of spins between the upper and lower states is not changed. T_{2} will be affected because phase coherence of the transverse magnetization is lost during exchange.
Another form of exchange is called chemical exchange. In chemical exchange, the A and B nuclei are from different molecules. Consider the chemical exchange between water and ethanol.
CH_{3}CH_{2}OH_{A} + HOH_{B} CH_{3}CH_{2}OH_{B} + HOH_{A}
Here the B hydrogen of water ends up on ethanol, and the A hydrogen on ethanol ends up on water in the forward reaction. There are four senarios for the nuclear spin, represented by the four equations.
Chemical exchange will affect both T_{1} and T_{2}. T_{1} is now affected because energy is transferred from one nucleus to another. For example, if there are more nuclei in the upper state of A, and a normal Boltzmann distribution in B, exchange will force the excess energy from A into B. The effect will make T_{1} appear smaller. T_{2} is effected because phase coherence of the transverse magnetization is not preserved during chemical exchange.
Bloch Equations
The Bloch equations are a set of coupled differential equations which can be used to describe the behavior of a magnetizatiion vector under any conditions. When properly integrated, the Bloch equations will yield the X', Y', and Z components of magnetization as a function of time.
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