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

964 Magnetic Resonance in Medicine

moment properties of several nuclei of current or poten- large magnetic moment, and for a rather intense field of
tial medical interest. Note that most of the nuclei in this 1.5T , this formula shows that, for every 10 million protons
table have a positive magnetic moment that corresponds in a sample, there are only 50 more nuclei in the favored,
to a spinning positive charge. Magnetic moments with a parallel state than in the higher energy, antiparallel state.
negative sign correspond to a spinning negative charge. If Other things being equal, it would improve the sensitivity
a magnetic field B is applied to the sample, the nuclei will of MRI if this population difference could be increased.
be given an energy −m · B. Thus, the energy of a state Equation (1) shows that this difference can be increased
depends on the orientation of m, and therefore J, with by increasing the field strength or by lowering the tem-
respect to the applied field. perature. It is clearly impractical to achieve a significant
The projection of the vector m in the direction of B must decrease in patient temperature, however, and there are
take one of the 2I + 1 values −γ hI , −γ h(I − 1), up to substantial technical difficulties in achieving body-sized
γ hI. The nucleus, therefore, has available to it 2I + 1 magnets much stronger than those currently in use.
states with different energies. The energy of these states When a material is magnetized, the strength and di-
will be equally spaced from one another by an amount rection of the effect is given by the vector M, called the
E = γ hB. The state with the lowest energy (the most magnetization, which is defined as the total magnetic mo-
favored state) has the magnetic moment most nearly par- ment per unit volume. If a region of volume V contains
allel to the applied field and vice versa. a large number of individual magnetic moments m i , then

Quantum mechanics predicts that if an oscillating mag- M = m i /V , where the sum is over all the sources in the
netic field is applied to the spin system, there will be a region.Thedimensionsofmareamperestimessquareme-
2
resonant exchange of energy between the field and the ters (A m ); therefore, the dimensions of M are amperes
spins when the quantum energy hω in the oscillating field per meter. If, as in the present case, the magnetization
corresponds to the separation between adjacent energy is proportional to the applied field, the susceptibility χ
levels. Transitions between nonadjacent energy levels are (which is dimensionless) is defined by the formula
not allowed. This criterion predicts a resonant interaction
when hω 0 = γ hB or, equivalently, ω 0 = γB. It is impor- M = χB/µ 0 , (2)
tant to note that this resonant frequency does not depend −7
where µ 0 = 4π × 10 H/m is a constant called the per-
on either h or I. This ties in with the fact that the nonquan-
meability of free space. For any material the total suscep-
tum, classical analysis, to be discussed later, of magnetic
tibility will be the sum of the contributions from each of
moments in a magnetic field gives the same value for the
the relevant sources of magnetic moment; the orbital elec-
characteristic frequency.
tron motion, the electron spin, and the nuclear spin. In the
The states with the magnetic moment in the direction of
present case, of course, we are particularly interested in
the field have a lower energy than those with the opposite
χ n , the contribution of the nuclei to the total susceptibility.
orientation. Consequently, if the spin system can come to
Statistical analysis of the distribution of the nuclei among
equilibrium with its surroundings at a temperature T , the
the available energy states shows that a nucleus with a
lower energy states will become more populated than the 2
spin I, magnetic moment m (A m ), and a density of
higher energy states and the substance as a whole will take
ρ (spins per cubic meter) will have a nuclear magnetic
on a net nuclear magnetization. This represents an aggre-
susceptibility given by
gate effect of the tendency of all of the individual nuclei
to orient themselves parallel to the applied magnetic field. µ 0 ρm 2
χ n = . (3)
This tendency toward alignment is, of course, opposed by 3kT
the randomizing effects of the thermal energy present in
Pure water has a density ρ of 55 moles/liter, or equiva-
the material. 28 3
lently, 6.62 × 10 protons/m . Using the values in Table I,
The difference between energy levels caused by the ap-
the nuclear magnetic susceptibility for the protons in wa-
plied magnetic field is quite small in comparison to the −9
1
thermal energy. For a system with only two levels (I = ), ter is found to be 3.86 × 10 . The magnetic behavior of
2 water is particularly relevant to MRI because most of the
the ratio of the number of spins in the lower energy state
signal derived from human tissues originates from water
n + to that in the higher energy state n − is given by
molecules. Note that m in Table I and Eq. (3) refers to
E/kT γ hB /kT
n + /n − = e = e . (1) the total magnitude of the magnetic moment vector. An-
other convention often used is to refer to the maximum
Here, k is Boltzmann’s constant and T is the absolute observable component of the magnetic moment vector as
◦
temperature. At body temperature 37 C or 310 K, kT = the magnetic moment. In the notation used here this com-
√
4.28 × 10 −21 J. Even for protons, which have a relatively ponent is equal to m I/(I + 1).