42.7.3 Magnetic Resonance Imaging
The proton, like the electron, has an inherent angular momentum (spin) and an inherent magnetic moment. You can think of the proton as being like a little compass needle that can be in one of two positions, the positions we call spin-up and spin-down.
Radiation therapy is a beneficial use of nuclear physics.
A compass needle aligns itself with an external magnetic field. This is the needle’s lowest-energy position. Turning a compass needle by hand is like rolling a ball uphill; you’re giving it energy, but, like the ball rolling downhill, it will realign itself with the lowest-energy position when you remove your finger. There is, however, an unstable equilibrium position, like a ball at the top of a hill, in which the needle is anti-aligned with the field. The slightest jostle will cause it to flip around, but the needle will be steady in its upside-down configuration if you can balance it perfectly.
A proton in a magnetic field behaves similarly, but with a major difference: Because the proton’s energy is quantized, the proton cannot assume an intermediate position. It’s either aligned with the magnetic field (the spin-up orientation) or anti-aligned (spin-down). FIGURE 42.26a shows these two quantum states. Turning on a magnetic field lowers the energy of a spin-up proton and increases the energy of an anti-aligned, spin-down proton. In other words, the magnetic field creates an energy difference between these states.
FIGURE 42.26 Nuclear magnetic resonance is possible because spin-up and spin-down protons have slightly different energies in a magnetic field.
The energy difference is very tiny, only about . Nonetheless, photons whose energy matches the energy difference cause the protons to move back and forth between these two energy levels as the photons are absorbed and emitted. In effect, the photons are causing the protons’ spins to flip back and forth rapidly. The photon frequency, which depends on the magnetic field strength, is typically about 100 MHz, similar to FM radio frequencies.
FIGURE 42.26b shows how this behavior is put to use. A sample containing protons is placed in a magnetic field. A coil is wrapped around the sample, and a variable-frequency AC source drives a current through this coil. The protons absorb power from the coil when its frequency is just right to flip the spin back and forth; otherwise, no power is absorbed. A resonance is seen by scanning the coil through a small range of frequencies.
This technique of observing the spin flip of nuclei (the technique also works for nuclei other than hydrogen) in a magnetic field is called nuclear magnetic resonance, or nmr. It has many applications in physics, chemistry, and materials science. Its medical use exploits the fact that tissue is mostly water, and two out of the three nuclei in a water molecule are protons. Thus the human body is basically a sample of protons, with the proton density varying as the tissue density varies.
The medical procedure known as magnetic resonance imaging, or MRI, places the patient in a spatially varying magnetic field. The variations in the field cause the proton absorption frequency to vary from point to point. From the known shape of the field and measurements of the frequencies that are absorbed, and how strongly, sophisticated computer software can transform the raw data into detailed images such as the one shown in FIGURE 42.27.
FIGURE 42.27 Magnetic resonance imaging shows internal organs in exquisite detail.
As an interesting footnote, the technique was still being called nuclear magnetic resonance when it was first introduced into medicine. Unfortunately, doctors soon found that many patients were afraid of it because of the word “nuclear.” Hence the alternative term “magnetic resonance imaging” was coined. It is true that the public perception of nuclear technology is not always positive, but equally true that nuclear physics has made many important and beneficial contributions to society.
CHAPTER 42 CHALLENGE EXAMPLE A radioactive tracer
An 85 kg patient swallows a beta emitter that is to be used as a tracer. The isotope’s half-life is 5.0 days. The average energy of the beta particles is 0.35 MeV, and they have an RBE (relative biological effectiveness) of 1. Ninety percent of the beta particles are stopped inside the patient’s body and 10% escape. What total dose equivalent does this patient receive?
MODEL Beta radiation penetrates the body—enough that 10% of the particles escape—so this is a whole-body exposure. Even the escaping particles probably deposit some energy in the body, but we’ll assume that the dose is from only those particles that stop inside the body.
SOLVE The dose equivalent is the absorbed dose in Gy multiplied by the RBE of 1. The absorbed dose is the energy absorbed per kilogram of tissue, so we need to find the total energy absorbed from the time the patient swallows the emitter until it has all decayed. The sample’s initial activity is related to the nuclear lifetime and the initial number of radioactive atoms by . Thus the number of radioactive atoms in the sample, all of which are going to decay and emit a beta particle, is
In developing this relationship, we used the connection between the lifetime and the half-life.
The initial activity is given in microcuries. Converting to becquerels, we have
The half-life in seconds is
Thus the total number of beta decays over the course of several weeks, as the sample completely decays, is
Ninety percent of these decays deposit, on average, 0.35 in the body, so the absorbed energy is
This is not a lot of energy in an absolute sense, but it is all damaging, ionizing radiation. The absorbed dose is
and thus the dose equivalent is
REVIEW This dose, typical of many medical uses of radiation, is about 15% of the yearly radiation dose from the natural background. Although one should always avoid unnecessary radiation, this dose would not cause concern if there were a medical reason for it.