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

μHere, μ_{s}= g_{s}μ_{B}m_{s}.

μwhere m_{B}= e h / 4 π m_{e}= 9.274 * 10^{-24}Am^{2},

This gives the electron a magnetic moment of -9.28476 * 10^{-24} Am^{2}. The magnetic moment of the proton is 1.4106 * 10^{-26} Am^{2}. 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 m_{s} 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} Am^{2}. 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**.

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

g μHere 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_{N}I.

μwhere m_{N}= e h / 4 π m_{p}= 5.0501 * 10^{-27}Am^{2},

{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

-g μand_{N}I B

+g μSuppose 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._{N}I B.

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:

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 μThe process we have just described is the essence of Magnetic Resonance Imaging (MRI), in which we:_{N}I B |.

- Orient all the nuclear spins in the object (ie., a patient's body) in parallel with a strong magnetic field.
- Flip the spins of the nuclei we are interested in locating in the other direction with a strong pulse of radiation of exactly the right frequency.
- Listen for the electromagnetic signal (the radiated photons) when the spins relax to their original state; the frequency will identify the isotope.

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.

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