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Encyclopedia of Physical Science and Technology EN008M-395 June 29, 2001 15:52

966 Magnetic Resonance in Medicine

which normally last a few milliseconds or less. When- relaxation processes to be described later. These latter ef-
ever Eq. (4) is valid, vector calculus states that the time fects normally produce changes in M that occur on the
rate of change of M must be perpendicular to M because millisecond to seconds time scale.
of the properties of the vector cross product. This means The most common approach to creating the magnetic
that, under the action of the externally applied fields, M resonance phenomena is to use a strong field, which we
cannot change in length. Therefore, if by some means, a shall designate as B 0 , in the z direction and to add to
nuclear magnetization has been created within a specimen it a weaker oscillating field B 1 that is oriented at right
at an initial time, the externally applied forces will cause angles to the z axis. If the oscillating field has both x
the magnetization to move with time, but only in a way and y components, B = B 1 (cos ω 1 tˆ ı − sin ω 1 tˆ ), then in
that keeps the length of the magnetization vector constant. a frame rotating at the frequency ω 1 (moving clockwise
The simplest situation is when the external field is simply when viewed from the positive z direction) it is just a con-
a constant B directed along the z axis. If at t = 0 there stant B 1 along the rotating x direction. Because such a
is a transverse component of M, M t , pointing along the field contains two components at right angles to one an-
x axis, the solutions to Eq. (4) are other, it is referred as a quadrature excitation field and is
said to be circularly polarized with a clockwise rotation.
M x = M t cos ω 0 t and M y =−M t sin ω 0 t,
It is not hard to show that a linearly polarized field with
where ω 0 = γB and M z is constant. That is, the component twice the amplitude (e.g., B = 2B 1 cos ω 1 tˆ ı) will have the
of the magnetization along the z axis remains unchanged same effect on the spins as the circularly polarized field
as time goes on, while the transverse magnetization rotates above. Interestingly a field rotating in the “wrong” direc-
at a rate (called the Larmor frequency) about the direction tion B = B 1 (cos ω 1 tˆ ı + sin ω 1 t, ˆ ) will have essentially no
of the applied magnetic field. Thus the total M vector at effect on the spins. In this article we will assume that a
any point moves steadily, at a constant rate, sweeping out a quadrature B 1 is used since this simplifies the analysis
cone whose axis is the direction of B. This motion is analo- somewhat.
gous to the motion of a rapidly spinning gyroscope (such The exact solution of Eq. (4) in these circumstances
as a top) responding to its own weight. This precession is not difficult but the results are more complex than we
is of basic importance to the detection of nuclear mag- wish to present here. The essential features of the solution
netism that, as has been previously mentioned, is much are that the B 1 field has a negligible effect on the mo-
too weak to be detected directly by the magnetic forces tion of M unless its frequency ω 1 is close to the Larmor
it exerts. The precession of the transverse magnetization frequency ω 0 . More specifically, unless ω 1 is within a fre-
produces a time-dependent magnetic field and, therefore, quency range γB 1 of ω 0 , the oscillating field will be in-
by Faraday’s law, a time-dependent electric field. This effective. If ω 1 is equal to ω 0 , the motion in the rotating
electric field can be detected as a voltage in a coil situated coordinate system is very simple. Then the magnetization
outside the sample. It is of fundamental importance that vector will rotate about the B 1 field, which will be constant
this induced voltage turns out to be large enough, in many in this frame at the rate γB 1 . Thus, if the magnetization is
cases, that the precessing nuclear magnetization can be along the z axis at time t = 0, it will rotate about B 1 and
detected electronically. Note that, although different lines will make an angle θ = γB 1 t with the z axis after time t.
of reasoning are used, both the quantum mechanical and Thus, if the oscillating field operates for a time t equal
◦
the macroscopic approaches lead to a characteristic fre- to π/(2γB 1 ), the magnetization will rotate 90 (π/2 rad)
quency given by ω 0 = γB. and will be located in the transverse plane. If it operates
A concept that is often utilized to describe the motion of for twice this time, the magnetization will be rotated 180 ◦
the magnetization vector is a coordinate system rotating at and would be inverted from its initial position.
or near the Larmor frequency. If the new coordinate sys- If a sample is placed in a strong magnetic field B 0 , the
tem is rotating at exactly the Larmor frequency, then in it, earlier analysis shows that initially the magnetization is
for the example previously given, there is no motion of the zero, but should increase with time to an equilibrium value
magnetization vector. To specify the direction of rotation, M 0 = χ n B 0 /µ 0 .Thegyroscopicequationscannotdescribe
we note that nuclei with positive values for the gyromag- this process since Eq. (4) shows that the length of the M
netic ratio precess in a clockwise direction when viewed vector cannot be changed by the external fields. The inter-
from the positive z direction. The advantage of the rotat- nal fields provide the answer to this paradox as they permit
ing coordinate system is that it disentangles the very rapid an exchange of energy between the surroundings, referred
precession motion, which usually takes place at mega- to somewhat loosely as the lattice, and the nuclear spin
hertz rates, and that is caused by the strong static field, system. Because the internal fields are the result of rapid,
from the much slower motions produced by weak, super- essentially chaotic, motion of the atoms of a liquid relative
imposed, oscillating external magnetic fields and from the to one another, it is extremely difficult to calculate, from