Friday, May 25, 2012

The Semiempirical Mass Formula

When revising Chapter 17 about Nuclear Medicine for the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I were tempted to include a discussion of the semiempirical mass formula, one of the fundamental concepts in nuclear physics. We finally decided that you cannot discuss everything in one book, but we did include the following footnote:
“This parabola and the general behavior of the binding energy with Z and A can be explained remarkably well by the semiempirical mass formula [Evans (1955, Chapter 11); Eisberg and Resnick, (1985, p. 528)].”
The semiempirical mass formula consists of five terms, which together predict the binding energy of a nucleus having atomic number Z and mass number A.
  1. The first term is negative, and arises from the binding caused by the short range nuclear force. It is proportional to A, which implies that it increases with the volume of the nucleus (this term assumes that the nuclear density is constant; the “liquid drop model”).
  2. The second term represents a positive correction caused by surface tension, arising because nucleons at the surface of the nucleus feel an attractive force from only one side (the nuclear interior). It is proportional to surface area, or A2/3.
  3. All the positively charged protons repel each other, and this effect is accounted for by a positive term for the Coulomb energy, proportional to Z2/A1/3.
  4. Everything else being equal, nuclei tend to be more stable if they have the same number of protons and neutrons. This behavior is reflected in an asymmetry term containing (Z - A/2)2/A. It is zero if A = 2Z (an equal number of protons and neutrons) and is positive otherwise.
  5. Finally, a pairing term is negative if both the number of protons and neutrons is even, positive if both are odd, and zero if one is even and the other odd.
The sum of these five terms is the semiempirical mass formula, with the terms weighted by parameters determined by fitting the model to data.

What can this formula explain? One example is the plot of average binding energy per nucleon as a function of A given in Fig. 17.3. At low A, this function predicts a very low binding energy because of the surface term (very small nuclei have a large surface-to-volume ratio). As A increases, the surface term becomes less important, but the Coulomb term increases as the nucleus is packed with more and more positive charge. For nuclei above about A = 60, the Coulomb term causes the binding energy to decrease as A increases. Therefore, the binding energy per nucleon reaches a peak for isotopes of elements such as iron and nickel, the most stable of nuclei, because of a competition between the surface and Coulomb terms. Although Russ and I did not mention it in our book, the smooth curve that most of the data cluster about in Fig. 17.3 is the prediction of the semiempirical mass formula.

If you hold A constant, you can examine the binding energy as a function of Z. This case is important for beta decay (in which a neutron is converted to a proton and an electron) and positron decay (in which a proton is converted to a neutron and a positron). The two terms in the semiempirical mass formula containing Z—the Coulomb term and the asymmetry term—combine to give a quadratic shape for the binding energy, as shown in Fig. 17.6. For odd A, the resulting parabola predicts the stable isotope (Z) for that A. For even A, the pairing term results in two parabolas, one for even Z and one for odd Z (Fig. 17.7).

In their textbook, Eisberg and Resnick conclude that
“The liquid drop model is the oldest, and most classical nuclear model. At the time the semiempirical mass formula was first developed, mass data was available, but not much else was known about nuclei. The parameters were purely empirical, and there was not even a qualitative understanding of the asymmetry and pairing terms. Nevertheless, the formula was significant because it described fairly accurately the masses of hundreds of nuclei in terms of only five parameters.”

Friday, May 18, 2012

Spin Echo

The spin-echo of nuclear magnetic resonance is one of those concepts that anyone interested in medical physics should know. Russ Hobbie and I discuss the spin-echo’s role in magnetic resonance imaging in Chapter 18 of the 4th edition of Intermediate Physics for Medicine and Biology.
“The pulse sequence shown in Fig. 18.17 can be used to determine T2 [the true or non-recoverable spin-spin relaxation time] and T*2 [the experimental spin-spin relaxation time]. Initially a π/2 [90°] pulse nutates M [the magnetization] about the x’ axis so that all spins lie along the rotating y’ axis. Figure 18.17(a) shows two such spins. Spin a continues to precess at the same frequency as the rotating coordinate system; spin b is subject to a slightly smaller magnetic field and precesses at a slightly lower frequency, so that at time TE/2 it has moved clockwise in the rotating frame by angle θ, as sown in Fig. 18.17(b). At this time a π [180°] pulse is applied that rotates all spins around the x’ axis. Spin a then points along the –y’ axis; spin b rotates to the angle shown in Fig. 18.17(c). If spin b still experiences the smaller magnetic field, it continues to precess clockwise in the rotating frame. At time TE both spins are in phase again, pointing along –y’ as shown in Fig. 18.17(d). The resulting signal is called an echo, and the process for producing it is called a spin-echo sequence.”
When I discuss this concept in class, I use the analogy of a footrace. Suppose all runners line up at the starting line, and at the sound of the starter's gun they begin to run clockwise around a track. Because they all run at somewhat different speeds, the pack of runners spreads until eventually (after many laps) they are distributed nearly evenly, and seemingly randomly, around the track. At this time another gun is fired, commanding all runners to turn around and run counterclockwise. Now, the fast runners who were ahead of the others are suddenly behind, and the slow runners who were behind the others are miraculously ahead. As time goes on, the fast runners catch up to the slow ones, and eventually they all meet in one tight pack as they run past the starting line. This unexpected regrouping of the runners is the echo. The analogy is not perfect, because the spins always precess in the same direction. Nevertheless, the 180° pulse has the effect of placing the fast spinners behind the slow spinners, which is the essence of both the spin echo effect and the runner analogy.

The spin-echo was first observed by physicist Erwin Hahn. His paper Spin Echos (Physical Review, 80:580-594, 1950) has been cited over 3000 times. Hahn wrote a “citation classic” article about this paper, in which he describes how he made his great discovery by accident.
“One day a strange signal appeared on the oscilloscope, in the absence of a pulse pedestal, so I kicked the apparatus and breathed a sigh of relief when the signal went away. A week later, the signal returned, and this time it checked out to be a real spontaneous spin echo nuclear signal from the test sample of protons in the glycerine being used. In about three weeks, I was able to predict mathematically what I suspected to be a constructive interference of precessing nuclear magnetism components by solving the Bloch nuclear induction equations. Here for the first time, a free precession signal in the absence of driving radiation was observed first, and predicted later. The spin echo began to yield information about the local atomic environment in terms of various amplitude and frequency memory beat effects, certainly not all understood in the beginning.

As I look back at this experience, it was an awesome adventure to be alone with the apparatus showing one new effect after another at a time when there was no one at Illinois experienced in NMR with whom I could talk."
You can learn more about Hahn and his discovery of the spin-echo from the transcript of an oral history interview published by the Niels Bohr Library and Archives, part of the American Institute of Physics.

For those of you who are visual learners, wikipedia has a nice animation of the formation of a spin-echo. Another animation is at http://mrsrl.stanford.edu/~brian/mri-movies/spinecho.mpg.

You can find an excellent video about spin-echo NMR on youtube, narrated by Sir Paul Callaghan, a New Zealand physicist (this is part of a series of videos that nicely support the discussion in Chapter 18 of Intermediate Physics for Medicine and Biology). Callaghan was a leader in MRI physics, and wrote Principles of Nuclear Magnetic Resonance Microscopy and, more recently, Translational Dynamics and Magnetic Resonance: Principles of Pulsed Gradient Spin Echo NMR. Tragically, Callaghan lost his battle to colon cancer this March.

Friday, May 11, 2012

Stopping Power and the Bragg Peak

Proton therapy is becoming a popular treatment for cancer. Russ Hobbie and I discuss proton therapy in Chapter 16 of the 4th edition of Intermediate Physics for Medicine and Biology.
Protons are also used to treat tumors. Their advantage is the increase of stopping power at low energies. It is possible to make them come to rest in the tissue to be destroyed, with an enhanced dose relative to intervening tissue and almost no dose distally (“downstream”) as shown by the Bragg peak in Fig. 16.51. … The edges of proton fields are much sharper than for x rays and electrons. This can provide better tissue sparing, but it also means that alignments must be much more precise [Moyers (2003)]. Sparing tissue reduces side effects immediately after treatment. It also reduces the incidence of radiation-induced second cancers many years later.”
Stopping power and range are a key concepts in describing how radiation interacts with matter, and are defined in Chapter 15.
“It is convenient to speak of how much energy the charged particle loses per unit path length, the stopping power, and its range—roughly, the total distance it travels before losing all its energy. The stopping power is the expectation value of the amount of kinetic energy T lost by the projectile per unit path length. (The term power is historical. The units of stopping power are J m−1 not J s−1.)”
To illustrate these concepts, I have devised a new homework problem. It is a bit like Problem 31 in Chapter 16, but uses a simpler expression for the energy dependence of the stopping power, and focuses on how this leads to a Bragg peak. This problem occasionally appears on the qualifier exam taken by our Medical Physics students at Oakland University.

Section 16.11

Problem 31 ½   Assume the stopping power of a particle, S = − dT/dx, as a function of kinetic energy, T, is S = C/T. 
(a) What are the units of C? 
(b) If the initial kinetic energy at x = 0 is To, find T(x) .
(c) Determine the range R of the particle as a function of C and To
(d) Plot S(x) versus x. Does this plot contain a Bragg peak? 
(e) Discuss the implications of the shape of S(x) for radiation treatment using this particle.

The stopping power often does fall as 1/T for large energies, as assumed in the above problem, but it rises as the square root of T for small energies (See Fig. 15.17 in Intermediate Physics for Medicine and Biology). To find a more accurate expression for S(x), try repeating this problem with

S(T) = C/(T + A/√T) .

Warning: I was not able to find a simple analytical expression for S(x) in this case. Can you?

One can imagine a proton incident with such low energy that it lies entirely on the rising part of the stopping power versus energy curve. In that case, a good approximation for the stopping power would be simply

S(T) = B √T .

I was able to solve for the stopping power in this case, although the expression is cumbersome. Interestingly, for these low energy particles the range is now infinite, because as the particle slows down it loses energy more slowly. I suppose once the particle’s energy is similar to the thermal energy, the entire model breaks down, so I am not too worried about this result.

These considerations illustrate how we gain much insight by examining simple toy models. That tends to be the view Russ and I adopt in our book, which is at odds with the traditional view of biologists and medical doctors, who relish the diversity and complexity of life.

Friday, May 4, 2012

The Optics of Life

As I mentioned two weeks ago, I’ve been reading The Optics of Life: A Biologist’s Guide to Light in Nature, by Sonke Johnsen. The book is delightful, exploring the biological implications of many fascinating phenomena such as scattering, interference, fluorescence, and bioluminescence. To me, The Optics of Life does for light what Life in Moving Fluids does for fluid dynamics; it explains how basic principles of physics apply to the diversity of life. Today, I want to focus on Chapter 8 of Johnsen’s book, about polarization.

The polarization of light is one of those topics Russ Hobbie and I don’t cover in the 4th edition of Intermediate Physics for Medicine and Biology. We only hint at its importance in Chapter 14 about Atoms and Light, when discussing Terahertz radiation.
“Classical electromagnetic wave theory is needed to describe the interactions [of Terahertz radiation with the body], and polarization (the orientation of the E vector of the propagating wave) is often important.”
Had you asked me two weeks ago why Russ and I skipped polarization, I would have said “because there are so few biological applications”. Johnsen proves me wrong. He writes
“As I mentioned earlier, aside from the few people who can see Haidinger’s Brush in the sky, the polarization characteristics of light are invisible to humans. However, a host of animals can detect one or both aspects of linearly polarized light (see Talbot Waterman’s massive review [1981] and Horvath and Varju’s even more massive book [2004] for comprehensive lists of taxa). Arthropods are the big winners here, especially insects, though also most crustaceans and certain spiders and scorpions. In fact, it is unusual to find an insect without polarization sensitivity. Outside the arthropods, the other major polarization-sensitive invertebrates are the cephalopods. Among vertebrates, polarization sensitivity is rarer and more controversial, but has been found in certain fish (primarily trout and Talbot salmon), some migratory birds and a handful of amphibians and reptiles. It is important to realize, though, that there is a serious sampling bias. Testing for polarization sensitivity is difficult and so has usually only been looked for in migratory animals and those known to be especially good at navigation, such as certain desert ants. The true prevalence of polarization sensitivity is unknown.

The ability to sense the polarization of light has been divided into two types. One is known as ‘polarization sensitivity.’ Animals that have polarization sensitivity are not much different from humans wearing Polaroid sunglasses. Polarization affects the intensity of what they see—but without a lot of head-turning and thinking, they cannot reliably determine the angle or degree of polarization or even separate polarization from brightness. The other type is known as ‘polarization vision.’ Animals with polarization vision perceive the angle and degree of polarization as something separate from simple brightness differences. Behaviorally, this means that they can distinguish two lights with different degrees and/or angles of polarization regardless of their relative radiances and colors. This is much like the definition of color vision, which involves the ability to distinguish two lights of differing hue and/or saturation regardless of their relative radiances.”
How I would love to have polarization vision! It would be an entirely new sensory experience. When Dorothy entered the land of Oz, she went from a black and white world to the richness of color. Now imagine a similar experience when going from our drab nonpolarized vision to polarization vision; it would offer a whole new way to view the world; a sixth sense. Alas, not all animals have polarization sensitivity, and even fewer have polarization vision. How these senses work is still unclear.
“While polarization sensitivity is certainly rarer among vertebrates [than invertebrates], it does exist… The mechanism of polarization sensitivity in vertebrates remains—along with the basis of animal magnetoreception—one of the two holy grails of sensory biology.”
My favorite example discussed by Johnsen is the Mantis shrimp, which can distinguish between left-handed and right-handed circularly polarized light. They do this by passing the light through a biological quarter-wave plate. The quarter-wave plate was one of my favorite topics in my undergraduate optics class. Incoming linearly polarized light is converted into circularly polarizing light by inducing a phase difference of 90 degrees between the two linear components. Similarly, the plate can convert circularly polarized light into linearly polarized light. Circularly polarized light always struck me as somehow magical. You can’t detect it using a sheet of plastic polarizing film, yet it is as fundamental a polarization state for light as is linear polarization. That the Mantis shrimp could make use of a quarter-wave plate to detect circularly polarized light is truly awesome.

Let me conclude by quoting the first sentence of Johnsen’s introduction, which to me elegantly sums up the book itself.
“Of all the remarkable substances of our experience—rain, leaves, baby toes—light is perhaps the most miraculous.”

Added note in the evening of May 4: Russ Hobbie reminds me that on the book's website is text from the first edition of Intermediate Physics for Medicine and Biology about optics, including much about polarization!