We have already observed that electric charge is quantized in units of e, as manifested in the cases of the electron and the proton. These objects also have an intrinsic angular momentum, which we call spin. Spin is also quantized: it is only observed in integer multiples of h / 2, where h is Planck's Constant (equal to 6.626 x 10-34 J s; notice that these units are indeed units of angular momentum).
The fact that these objects have angular momentum indicates that they are not point particles. A point particle has no dimension: no length, width or breadth. It therefore has zero lever arm relative to its center and cannot have angular momentum. We do not have an intuitive model for electrons and protons (or any other of the so-call elementary particles), but they are most definitely not pointlike. Yet they seem that way at every length scale larger than atomic because they are very small. And since they are rotating, charged objects, they act as tiny current loops. Therefore they have magnetic moments.
We can measure the magnetic moment of an electron by measuring its potential energy in a magnetic field. The result is expressed as
μs = gs μB ms.Here, μs is called the spin magnetic moment, gs is the spin gyromagnetic ratio, μB is the Bohr magneton and ms is 1/2 or -1/2 (the spin of the electron divided by h). Of these numbers, only the Bohr magneton has physical units. Its value is
μB = e h / 4 π mewhere me is the mass of an electron. If we take e, h and me to be natural constants, the measurement of μs is actually a measurement of gs: its value is -2.0023193043617 (B. Odom et al., Physical Review Letters 97, 030801, 2006). This value has a relative error of 7.6 x 10-13, making it the most accurate measurement in science. What is remarkable is that the theoretical prediction for this value is in total agreement!= 9.274 * 10-24 Am2,
This gives the electron a magnetic moment of -9.28476 * 10-24 Am2. The magnetic moment of the proton is 1.4106 * 10-26 Am2. These numbers, like their charges, masses and the allowed values of their spin, are universal in the sense that EVERY electron has the same values as every other electron, and EVERY proton has the same values as every other proton. Such values are often called quantum numbers, and the electrons and protons themselves are often called quanta. The number ms above is called the spin quantum number.
The nuclei of atoms contain protons and neutrons. Since a neutron is electrically neutral, you might expect it to have no magnetic moment. In fact, it has a magnetic moment of -9.6624 * 10-27 Am2. How can this be?
Protons and neutrons are made up of smaller elementary particles called quarks. The force which binds the quarks together is called the strong force. It acts like a spring whose spring constant gets stronger as the distance between the quarks increases, so they are never seen alone. The quarks come in six flavors, which have been dubbed up, down, charm, strange, top and bottom.
The proton is made of 2 up quarks and 1 down quark, and the neutron is made of 1 up quark and 2 down quarks. The up quarks have an electrical charge of 2e/3, while the down quarks have an electrical charge of -e/3. All have spin quantum numbers of 1/2 or -1/2. This means that while the neutron is electrically neutral, it still has spinning charges within, and hence can have a nonzero magnetic moment.
By the same token, the nucleus of all atoms have spin, since they are collections of spinning protons and neutrons. The nuclear magnetic moment of a particular atom is
The mechanism for these changes in energy is the photon: a quantum of electromagnetic radiation. Recall Huygens' Principle: a wave travels as if its energy is propagated from one oscillator to the next. The electromagnetic field is modeled using electromagnetic oscillators: at every point in space, there is an oscillator which is the source for an electric and a magnetic field. A photon is a single excitation moving from one oscillator to the next, and its energy is proportional to the frequency of oscillation:
To create an image, we use a magnetic field gradient: by creating a different value of the magnetic field at every point in the object, each point will be associated with a different frequency. Empirically, we find that different molecules have different relaxation times. By recording the timing of the radiated photons as well as their frequencies, the image can contain identifying information as well.
In T1-weighted images, tissue containing lipids (fat) appear light while tissues with mostly water appear
darker. For T2-weighted images, the reverse is true.
For any given tissue type, T1 times are generically longer than T2 times. By varying the pulse rate and
direction, an image can be formed from signals corresponding to one or the other, allowing tissue
differentiation.
Notice the small dark circles about mid-image at around 11:00-12:00 in each axial section. These are blood vessels. They appear dark because
the blood whose spins were flipped for this image has flowed out of the plane of the image.
The next applet provides you with the g factor and spin for a nucleus in a magnetic field, and asks you for the nuclear magnetic moment, the energy change as the spin is flipped and the resulting photon frequency upon relaxation. Assume that all spin flips are between the minimum and maximum energy levels.
The next section is about the structure of atoms.
©2007, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
Please send comments or suggestions to the author.
Nuclear Magnetic Moments
g μN I.
Here the gyromagnetic ratio has a different value for each atom, which depends not only on the species but on its immediate environment as well, and the nuclear magneton
μN = e h / 4 π mp
where mp is the mass of a proton. I is the nuclear spin; the spin quantum number for a nucleus can be any number in the set
= 5.0501 * 10-27 Am2,
{I, I - 1, I - 2, ..., -I + 2, -I + 1, -I}.
The magnetic potential energy of a nucleus in an external magnetic field is therefore quantized, and can have 2 I + 1 values, or energy levels, between
-g μN I B
and
+g μN I B.
Suppose we place an object in an intense magnetic field, on the order of 1 to 10 T: all of the nuclei in all of its atoms will orient themselves to be parallel with that field in order to minimize their potential energies. But if they were to absorb an amount of energy equal to the difference between the minimum and maximum energies, their spins would flip: they would now be pointing in the opposite direction to the magnetic field. Of course, each nucleus would very quickly lose that energy, since it is most stable at the lower energy, when its spin is parallel to the magnetic field.
E = h ν.
If the spins are flipped from their orientations of minimum energy to their orientations of maximum energy, the energy of the photon which flips the spin, as well as the energy of the photon lost when the spin flips back, is
| 2 g μN I B |.
The process we have just described is the essence of Magnetic Resonance Imaging (MRI), in which we:
MRI Examples
Diagnostic MRIs are typically acquired as both T1-weighted and T2-weighted images.
T1 corresponds to the relaxation time constant in the direction parallel to the ambient magnetic field.
T2 corresponds to the relaxation time constant in the plane perpendicular to the ambient magnetic field.
Series 1 - Sagittal T1 of the Lumbar Region (6 mm between sections)
Note that the spinal cord appears light in this series but dark in the next; it has a high lipid content.
Series 2 - Sagittal T2 of the Lumbar Region (6 mm between sections)
Series 3 - Axial T1 around L4 (5.6 mm between sections)
Series 4 - Axial T2 around L4 (5.6 mm between sections)