FIGURE 28.24 Energy levels for a proton in a magnetic field. Quantum mechanics limits the proton to two possible energies . . . ... which correspond to two possible orientations, aligned with or opposite the magnetic field. Energy E2 = +µB- 0- E = -µB- FIGURE 28.25 An MRI image shows the cross section of a patient's head.
The magnetism of permanent magnets arises because the inherent magnetic moment of electrons causes them to act like little compass needles. Protons also have an inherent magnetic moment, and this is the basis for magnetic resonance imaging (MRI) in medicine.
Although a compass needle would prefer to align with a magnetic field, the needle can point in any direction. This isn’t the case for the magnetic moment of a proton. Quantum physics tells us that the proton’s energy must be quantized. There are only two possible energy levels—and thus two possible orientations—for protons in a magnetic field:
E1 = -μB magnetic moment aligned with the field
E2 = +μB magnetic moment aligned opposite the field
where μ = 1.41 x 10-26 J/T is the known value of the proton’s magnetic moment. FIGURE 28.24 shows the two possible energy states. The magnetic moment, like a compass needle, “wants” to align with the field, so that is the lower-energy state. Human tissue is mostly water. Each water molecule has two hydrogen atoms whose nuclei are single protons. In a magnetic field, the protons go into one or the other quantum state. A photon of just the right energy can “flip” the orientation of a proton’s magnetic moment by causing a quantum jump from one state to the other. The energy difference between the states is small, so the relatively low-frequency photons are in the radio portion of the
probe coil that emits radio waves. When the probe is tuned to just the right frequency, the waves are in resonance with the energy levels of the protons, thus giving us the name magnetic resonance imaging.
The rate of absorption of these low-energy photons is proportional to the density of hydrogen atoms. Hydrogen density varies with tissue type, so an
MRI image—showing different tissues—is formed by measuring the variation across the body of the rate at which photons cause quantum jumps between the two proton energy levels. A figure showing the absorption rate versus position in the body is an image of a “slice” through a patient’s body, as in FIGURE 28.25.
a. An MRI patient is placed inside a solenoid that creates a strong magnetic field. If the field strength is 2.00 T, to what frequency must the probe coil be set? What is the wavelength of the photons produced?
b. In a uniform magnetic field, all protons in the body would absorb photons of the same frequency. To form an image of the body, the magnetic field is designed to vary from point to point in a known way. Because the field is different at each point in the body, each point has a unique frequency of photons that will be absorbed. The actual procedure is complex, but consider a simple model in which the field strength varies
only along the axis of the patient’s body, which we will call the x-axis. In particular, suppose that the magnetic field
strength in tesla is given by B = 2.00 + 1.60x, where x, measured from a known reference point, is in meters. The probe coil is first tuned to the resonance frequency at the reference point. As the frequency is increased, a strong signal is observed at a frequency 4.7 MHz above the starting frequency. What is the location in the body, relative to the reference
point, of the tissue creating this strong signal?
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