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 can’t 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 A^2/3.
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/A^1/3.
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).

Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles,
by Eisberg and Resnick.

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.

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 T_E/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 T_E 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, Volume 80, Pages 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.

Paul Callaghan discusses the spin echo.

https://www.youtube.com/watch?v=B2HMAJQJ7ok

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’s 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 graduate 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 T_o, find T(x) .
(c) Determine the range R of the particle as a function of C and T_o. 
(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 wasn’t 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.

The Optics of Life

The Optics of Life:
A Biologist's Guide to Light in Nature,
by Sonke Johnsen.

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’d 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 awesome.

Let me conclude by quoting the first sentence of Johnsen’s introduction to The Optics of Life, which 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!

Physics and Medicine

Readers of Intermediate Physics for Medicine and Biology already know how important physics is to medicine. Now, subscribers to the famed British medical journal The Lancet are learning this too. The April 21–27 issue (Volume 379, Issue 9825) of The Lancet contains a series of articles under the heading “Physics and Medicine.” In his editorial introducing this series, Peter Knight (president of the Institute of Physics) calls for UK medical schools to reinstate an undergraduate physics requirement for aspiring premed students. The English don’t require their premed students to take physics? Yikes!

Richard Horton, editor-in-chief of The Lancet, seconds this call for better physics education. He concludes that “Young physicists need to be nurtured to ensure a sustainable supply of talented scientists who can take advantage of the opportunities for health-related physics research in the future. Schools, indeed all of us interested in the future of health care, should declare and implement a passion for physics. Our Series is our commitment to do so.” Bravo! Below I reproduce the abstracts to the five articles in the Physics and Medicine series. In brackets I indicate the chapter or section in the 4th edition of Intermediate Physics for Medicine and Biology where a particular topic is discussed.

Physics and Medicine: a Historical Perspective
Stephen F Keevil

Nowadays, the term medical physics usually refers to the work of physicists employed in hospitals, who are concerned mainly with medical applications of radiation, diagnostic imaging, and clinical measurement. This involvement in clinical work began barely 100 years ago, but the relation between physics and medicine has a much longer history. In this report, I have traced this history from the earliest recorded period, when physical agents such as heat and light began to be used to diagnose and treat disease. Later, great polymaths such as Leonardo da Vinci and Alhazen used physical principles to begin the quest to understand the function of the body. After the scientific revolution in the 17th century, early medical physicists developed a purely mechanistic approach to physiology, whereas others applied ideas derived from physics in an effort to comprehend the nature of life itself. These early investigations led directly to the development of specialties such as electrophysiology [Chpts 6, 7], biomechanics [Secs 1.5–1.7] and ophthalmology [Sec 14.12]. Physics-based medical technology developed rapidly during the 19th century, but it was the revolutionary discoveries about radiation and radioactivity [Secs 17.2–17.4] at the end of the century that ushered in a new era of radiation-based medical diagnosis and treatment, thereby giving rise to the modern medical physics profession. Subsequent developments in imaging [Chpt 12] in particular have revolutionised the practice of medicine. We now stand on the brink of a new revolution in post-genomic personalised medicine, with physics-based techniques again at the forefront. As before, these techniques are often the unpredictable fruits of earlier investment in basic physics research.

Diagnostic Imaging
Peter Morris, Alan Perkins

Physical techniques have always had a key role in medicine, and the second half of the 20th century in particular saw a revolution in medical diagnostic techniques with the development of key imaging instruments: x-ray imaging [Chpt 16] and emission tomography [Secs 12.4–12.6] (nuclear imaging [Secs 17.12-17.13] and PET [Sec 17.14]), MRI [Chpt 18], and ultrasound [Chpt 13] These techniques use the full width of the electromagnetic spectrum [Sec 14.1], from gamma rays to radio waves, and sound [Secs 13.1–13.3]. In most cases, the development of a medical imaging device was opportunistic; many scientists in physics laboratories were experimenting with simple x-ray images within the first year of the discovery of such rays, the development of the cyclotron and later nuclear reactors created the opportunity for nuclear medicine, and one of the co-inventors of MRI was initially attempting to develop an alternative to x-ray diffraction for the analysis of crystal structures. What all these techniques have in common is the brilliant insight of a few pioneering physical scientists and engineers who had the tenacity to develop their inventions, followed by a series of technical innovations that enabled the full diagnostic potential of these instruments to be realised. In this report, we focus on the key part played by these scientists and engineers and the new imaging instruments and diagnostic procedures that they developed. By bringing the key developments and applications together we hope to show the true legacy of physics and engineering in diagnostic medicine.

The Importance of Physics to Progress in Medical Treatment
Andreas Melzer, Sandy Cochran, Paul Prentice, Michael P MacDonald, Zhigang Wang, Alfred Cuschieri

Physics in therapy is as diverse as it is substantial. In this review, we highlight the role of physics—occasionally transitioning into engineering—through discussion of several established and emerging treatments. We specifically address minimal access surgery, ultrasound [Sec 13.7], photonics [Chpt 14], and interventional MRI, identifying areas in which complementarity is being exploited. We also discuss some of the fundamental physical principles involved in the application of each treatment to medical practice.

Future Medicine Shaped by an Interdisciplinary New Biology
Paul O'Shea

The projected effects of the new biology on future medicine are described. The new biology is essentially the result of shifts in the way biological research has progressed over the past few years, mainly through the involvement of physical scientists and engineers in biological thinking and research with the establishment of new teams and task forces to address the new challenges in biology. Their contributions go well beyond the historical contributions of mathematics, physical sciences, and engineering to medical practice that were largely equipment oriented. Over the next generation, the entire fabric of the biosciences will change as research barriers between disciplines diminish and eventually cease to exist. The resulting effects are starting to be noticed in front-line medicine and the prospects for the future are immense and potentially society changing. The most likely disciplines to have early effects are outlined and form the main thrust of this paper, with speculation about other disciplines and emphasis that although physics-based and engineering-based biology will change future medicine, the physical sciences and engineering will also be changed by these developments. Essentially, physics is being redefined by the need to accommodate these new views of what constitutes biological systems and how they function.

The Importance of Quantitative Systemic Thinking in Medicine
Geoffrey B West

The study and practice of medicine could benefit from an enhanced engagement with the new perspectives provided by the emerging areas of complexity science [Secs 10.7-10.8] and systems biology. A more integrated, systemic approach is needed to fully understand the processes of health, disease, and dysfunction, and the many challenges in medical research and education. Integral to this approach is the search for a quantitative, predictive, multilevel, theoretical conceptual framework that both complements the present approaches and stimulates a more integrated research agenda that will lead to novel questions and experimental programmes. As examples, the importance of network structures and scaling laws [Sec 2.10] are discussed for the development of a broad, quantitative, mathematical understanding of issues that are important in health, including ageing and mortality, sleep, growth, circulatory systems [Sec 1.17], and drug doses [Sec 2.5]. A common theme is the importance of understanding the quantifiable determinants of the baseline scale of life, and developing corresponding parameters that define the average, idealised, healthy individual.

Frequency versus Wavelength

The Optics of Life:
A Biologist's Guide to Light in Nature,
by Sonke Johnsen.

I am currently reading The Optics of Life: A Biologist’s Guide to Light in Nature, by Sonke Johnsen. I hope to have more to say about this fascinating book when I finish it, but today I want to consider a point made in Chapter 2 (Units and Geometry), which addresses the tricky issue of measuring light intensity as a function of either frequency or wavelength. Johnsen favors using wavelength whenever possible.

However, one critical issue must be discussed before we put frequency away for good. It involves the fact that light spectra are histograms. Suppose you measure the spectrum of daylight, and that the value at 500 nm is 15 photons/cm²/s/nm. That doesn’t mean that there are 15 photons/cm²/s with a wavelength of exactly 500 nm. Instead, it means that, over a 1-nm-wide interval centered on a wavelength of 500 nm, you have 15 photons/cm²/s. The bins in a spectrum don’t have to be 1 nm wide, but they all must have the same width.

Let’s suppose all the bins are 1 nm wide and centered on whole numbers (i.e., one at 400 nm, one at 401 nm, etc.). What happens if we convert these wavelength values to their frequency counterparts? Let’s pick the wavelengths of two neighboring bins and call them λ₁ and λ₂. The corresponding frequencies ν₁ and ν₂ are equal to c/λ₁ and c/λ₂, where c is the speed of light. We know that λ₁−λ₂ equals 1 nm, but what does ν₁−ν₂ equal?

ν₁−ν₂ = … = −c/λ₁²

…So the width of the frequency bins depends on the wavelengths they correspond to, which means they won’t be equal! In fact, they are quite unequal. Bins at the red end of the spectrum (700 nm) are only about one-third as wide as bins at the blue end (400 nm). This means that a spectrum generated using bins with equal frequency intervals would look different from one with equal wavelength intervals. So which one is correct? Neither or both. The take-home message is that the shape of a spectrum depends on whether you have equal frequency bins or equal wavelength bins.

Johnsen goes on to note that the wavelength at which the spectrum is maximum depends on if you use equal frequency or equal wavelength bins. It does not make sense to say that the spectrum of, say, sunlight peaks at a particular wavelength, unless you specify the type of spectrum you are using. Furthermore, you cannot unambiguously say light is “white” (a uniform spectrum). White light using equal wavelength bins is not white using equal frequency bins. Fortunately, if you integrate the spectrum, you get the same value regardless of if you express it in terms of wavelength or frequency.

Russ Hobbie and I discuss this issue in Chapter 14 (Atoms and Light) of the 4th edition of Intermediate Physics for Medicine and Biology.

Early measurements of the radiation function were done with equipment that made measurements vs. wavelength. It is also possible to measure vs. frequency. To rewrite the radiation function in terms of frequency, let λ₁ and λ₂ =  λ₁ + dλ be two slightly different wavelengths, with power W_λ(λ, T) dλ emitted per unit surface area at wavelengths between λ₁ and λ₂. The same power must be emitted between frequencies ν₁ = c/λ₁ and ν₂ = c/λ₂:

W_ν(ν,T) dν = W_λ(λ,T) dλ .     (14.35)

Since ν = c/λ, dν/dλ = − c/λ², and

|dν| = + c λ² |dλ| .                 (14.36)

... This transformation is shown in Fig. 14.24. The amount of power per unit area radiated in the 0.5 μm interval between two of the vertical lines in the graph on the lower right is the area under the curve of W_λ between these lines. The graph on the upper right transforms to the corresponding frequency interval. The radiated power, which is the area under the W_ν curve between the corresponding frequency lines on the upper left, is the same. We will see this same transformation again when we deal with x rays. Note that the peaks of the two curves are at different frequencies or wavelengths.

Students who prefer visual explanations should see Fig. 14.24, which Russ drew. It is one of my favorite pictures in our book, and provides an illuminating comparison of the two spectra.

One detail I should mention: why in Eq. 14.36 do we use absolute values to eliminate the minus sign introduced by the derivative dν/dλ? Typically, when you integrate a spectrum, you start from the lower frequency and go to the higher frequency (say, zero to infinity), and you start from the shorter wavelength and go to the longer wavelength (again, zero to infinity). However, zero frequency corresponds to an infinite wavelength, and an infinite frequency corresponds to zero wavelength. So, really one case should be integrated forward (zero to infinity) and the other backwards (infinity to zero). If we keep the convention of always integrating from zero to infinity in both cases, we introduce an extra minus sign, which cancels the minus sign introduced by dν/dλ.

Sometimes it helps to have an elementary example to illustrate these ideas. Therefore, I have developed a new homework problem that introduces an extremely simple spectrum for which you can do the math fairly easily, thereby allowing you to focus on the physical interpretation. Enjoy.

Section 14.7

Problem 23 ½ Let W_ν(ν) = A ν (ν_ο - ν) for ν less than ν_ο, and W_ν(ν) = 0 otherwise.
(a) Plot W_ν(ν) versus ν.
(b) Calculate the frequency corresponding to the maximum of W_ν(ν), called ν_max.
(c) Let λ_ο = c/ν_ο and λ_max = c/ν_max. Write λ_max in terms of λ_ο.
(d) Integrate W_ν(ν) over all ν to find W_tot.
(e) Use Eqs. 14.35 and 14.36 to calculate W_λ(λ).
(f) Plot W_λ(λ) versus λ.
(g) Calculate the wavelength corresponding to the maximum of W_λ(λ), called λ^*_max, in terms of λ_ο.
(h) Compare λ_max and λ^*_max. Are they the same or different? If λ_ο is 400 nm, calculate λ_max and λ^*_max? What part of the electromagnetic spectrum is each of these in?
(i) Integrate W_λ(λ) over all λ to find W^*_tot. Compare W_tot and W^*_tot. Are they the same or different?

Stirling’s Formula!

Factorials are used in many branches of mathematics and physics, and particularly in statistical mechanics. One often needs the natural logarithm of a factorial, ln(n!). In Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I use Stirling’s approximation to compute ln(n!). We analyze this approximation in Appendix I.

There is a very useful approximation to the factorial, called Stirling’s approximation:

ln(n!) = n ln n – n .

To derive it, write ln(n!) as

ln(n!) = ln 1 + ln 2 + … + ln n = ∑ ln m

The sum is the same as the total area of the rectangles in Fig. I.1, where the height of each rectangle is ln m and the width of the base is one. The area of all the rectangles is approximately the area under the smooth curve, which is a plot of ln m. The area is approximately

∫₁ⁿln m dm = [m ln m – m]₁ⁿ = n ln n – n + 1.

This completes the proof.

David Mermin—one of my favorite writers among physicists—has much more to say about Stirling’s approximation in his American Journal of Physics article “Stirling’s Formula!” (leave it to Mermin to work an exclamation point into his title). He writes Stirling’s approximation as n! = √(2 π n) (n/e)ⁿ. Taking the natural logarithm of both sides gives ln(n!) = ln(2 π n)/2 + n ln n – n . For large n, the first term is small, and the result is the same as Russ and I present. I wonder what affect the first term has on the approximation? For small n, it makes a big difference! In Table I.1 of our textbook, we compute the accuracy of n ln n – n for n = 5. In that case, n! = 120, so ln(n!) = ln(120) = 4.7875 and 5 ln 5 – 5 = 3.047, giving a 36% error. But ln(10 π)/2 + 5 ln 5 – 5 = 4.7708, implying an error of 0.35 %, so Mermin’s formula is much better than ours. (I shouldn’t call it Mermin’s formula; I believe Stirling himself derived n! = √(2 π n) (n/e)ⁿ.)

Mermin doesn’t stop there. He analyzes the approximation in more detail, and eventually derives an exact formula for n! that looks like Stirling’s approximation given above, except multiplied by an infinite product. In the process, he looks at the approximation for the base of the natural logarithms, e, presented in Chapter 2 of Intermediate Physics for Medicine and Biology, e = (1 + 1/N)^N, and shows that a “spectacularly better” approximation for e is (1 + 1/N)^N+1/2. He then goes on to derive an improved approximation for n!, which is his expression for Stirling’s formula times e^(1/12n). Perhaps getting carried away, he then derives even better approximations.

All of this matters little in applications to statistical mechanics, where n is on the order of Avogadro’s number, in which case the first term in Stirling’s formula is utterly negligible. Nevertheless, I urge you to read Mermin’s paper, if only to enjoy the elegance of his writing. To learn more about Mermin’s views on writing physics, see his essay “Writing Physics.”

Stokes' Law

Stokes’ law appears in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology. Russ Hobbie and I write

For a Newtonian fluid … with viscosity η, one can show (although it requires some detailed calculation⁶) that the drag force on a spherical particle of radius a is given by

 F_drag = − β v = − 6 π η a v.

This equation is valid when the sphere is so large that there are many collisions of fluid molecules with it and when the velocity is low enough so that there is no turbulence. The result is called Stokes’ law.

Footnote 6 says “This is an approximate equation. See Barr (1931, p. 171).”

We can derive the form of Stokes’s law from dimensional reasoning. For a spherical particle of radius a in a fluid moving with speed v and having viscosity η, try to create a quantity having dimensions of force from some combination of a (meter), v (meter/second), and η (Newton second/meter²; see Sec. 1.14). The only way to do this is the combination η a v. You get the form of Stokes’ law, except for the dimensionless factor of 6π. Calculating the 6π was Stokes’ great accomplishment.

Boundary Layer Theory,
by Schlichting and Gersten.

In order to learn how Stokes obtained his complete solution, I turn to one of my favorite books on fluid dynamics: Boundary Layer Theory, by Hermann Schlichting. Consider a sphere of radius R placed into in an unbounded fluid moving with speed U_∞. Assume that the motion occurs at low Reynolds number (a “creeping motion”), so that inertial effects are negligible compared to viscous forces. The Navier-Stokes equation (see Problem 28 of Chapter 1 in Intermediate Physics for Medicine and Biology) reduces to ∇ p = μ ∇² v, where p is the pressure, μ the viscosity, and v the fluid speed. Assume further that the fluid is incompressible, so that div v = 0 (see Problem 35 of Chapter 1), and that far from the sphere the fluid speed is v = U_∞. Finally, assume no-slip boundary conditions at the sphere surface, so that v = 0 at r = R. At this point, let us hear the results in Schlichting’s own words (translated from the original German, of course).

The oldest known solution for a creeping motion was given by G. G. Stokes who investigated the case of parallel flow past a sphere [17]. The solution of eqns. (6.3) [Navier-Stokes equation] and (6.4) [div v = 0] for the case of a sphere of radius R, the centre of which coincides with the origin, and which is placed in a parallel stream of uniform velocity U_∞, Fig. 6.1, along the x-axis can be represented by the following equations for the pressure and velocity components [Eqs. 6.7, which are slightly too complicated to reproduce in this blog, but which involve no special functions or other higher mathematics]. . . The pressure distribution along a meridian of the sphere as well as along the axis of abscissae, x, is shown in Fig. 6.1 [a plot with a peak positive pressure at the upstream edge and a peak negative pressure at the downstream edge]. The shearing-stress distribution over the sphere can also be calculated from the above formula. It is found that the shearing stress has its largest value [at a point along the equator in the sphere center] . . . Integrating the pressure distribution and the shearing stress over the surface of the sphere we obtain the total drag D = 6 π μ R U_∞ This is the very well known Stokes equation for the drag of a sphere. It can be shown that one third of the drag is due to the pressure distribution and that the remaining two thirds are due to the existence of shear. . . the sphere drags with it a very wide layer of fluid which extends over about one diameter on both sides.

Reference 17 is to Stokes, G. G. (1851) “On the Effect of Internal Friction of Fluids on the Motion of Pendulums,” Transactions of the Cambridge Philosophical Society, Volume 9, Pages 8–106.

Schlichting goes on to analyze the flow around a sphere for high Reynolds number, which is particularly fascinating because in that case viscosity is negligible everywhere except near the sphere surface where the no-slip boundary condition holds. This results in a thin boundary layer forming at the sphere surface. In his introduction, Schlichting writes

In a paper on “Fluid Motion with Very Small Friction,” read before the Mathematical Congress in Heidelberg in 1904, L. Prandtl showed how it was possible to analyze viscous flows precisely in cases which had great practical importance. With the aid of theoretical considerations and several simple experiments, he proved that the flow about a solid body can be divided into two regions: a very thin layer in the neighbourhood of the body (boundary layer) where friction plays an essential part, and the remaining region outside this layer, where friction may be neglected.

The book that Russ and I cite in footnote 6 is A Monograph of Viscometry by Guy Barr (Oxford University Press, 1931). I obtained a yellowing and brittle copy of this book through interlibrary loan. It doesn’t describe the derivation of Stokes law in as much detail as Schlichting, but it does consider many corrections to the law, including Oseen’s correction (a first order correction when expanding the drag force in powers of the Reynold’s number), corrections for the effects of walls, consideration of the ends of tubes, and even the mutual effect of two spheres interacting. I found the following sentence, discussing cylinders as opposed to spheres, to be particularly interesting: “Stoke’s approximation leads to a curious paradox when his system of equations is applied to the movement of an infinite cylinder in an infinite medium, the only stable condition being that in which the whole of the fluid, even at infinity, moves with the same velocity as the cylinder.” You can’t derive a Stokes’ law in two dimensions.

While boundary layer theory and high Reynolds number flow is important for many engineering applications, much of biology takes place at low Reynolds number where Stokes’ law applies. (For more about life at low Reynolds number, see “Life at Low Reynolds Number” by Edward Purcell.)

Asimov's Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

Stokes’ life is described in Asimov's Biographical Encyclopedia of Science and Technology

Stokes, Sir George Gabriel
British mathematician and physicist
Born: Skreen, Sligo, Ireland, August 13, 1819
Died: Cambridge, England, February 1, 1903

Stokes was the youngest child of a clergyman. He graduated from Cambridge in 1841 at the head of his class in mathematics and his early promise was not belied. In 1849 he was appointed Lucasian professor of mathematics at Cambridge; in 1854, secretary of the Royal Society; and in 1885, president of the Royal Society. No one had held all three offices since Isaac Newton a century and a half before. Stokes’s vision is indicated by the fact that he was one of the first scientists to see the value of Joule’s work.

Between 1845 and 1850 Stokes worked on the theory of viscous fluids. He deduced an equation (Stokes’s law) that could be applied to the motion of a small sphere falling through a viscous medium to give its velocity under the influence of a given force, such as gravity. This equation could be used to explain the manner in which clouds float in air and waves subside in water. It could also be used in practical problems involving the resistance of water to ships moving through it. In fact such is the interconnectedness of science that six decades after Stokes’s law was announced, it was used for a purpose he could never have foreseen—to help determine the electric charge on a single electron in a famous experiment by Millikan. . .

iBioMagazine

I recently discovered iBioMagazine, which I highly recommend. The iBioMagazine website describes its goals.

iBioMagazine offers a collection of short (less than 15 min) talks that highlight the human side of research. iBioMagazine goes 'behind-the-scenes' of scientific discoveries, provides advice for young scientists, and explores how research is practiced in the life sciences. New topics will be covered in each quarterly issue. Subscribe to be notified when a new iBioMagazine is released.

Here are some of my favorites:

Bruce Alberts, Editor-in-Chief of Science magazine and coauthor of The Molecular Biology of the Cell, tells about how he learned from failure.

Former NIH director Harold Varmus explains why he became a scientist.

Young researchers participating in a summer course at the Marine Biological Laboratory at Woods Hole explain why they became scientists.

Hugh Huxley discusses his development of the sliding filament theory of muscle contraction. Of particular interest is that Huxley began his career as a physics student, and then changed to biology. Andrew Huxley (no relation), of Hodgkin and Huxley fame, independently developed a similar model.

Finally, readers of Intermediate Physics for Medicine and Biology should be sure to listen to Rob Phillips’ wonderful talk about the role of quantitative thinking and mathematical modeling in biology. Phillips is coauthor of the textbook Physical Biology of the Cell, which I have discussed earlier in this blog.

Saltatory Conduction

Action potential propagation along a myelinated nerve axon is often said to occur by “saltatory conduction.” The 4th edition of Intermediate Physics for Medicine and Biology follows this traditional explanation.

We have so far been discussing fibers without the thick myelin sheath. Unmyelinated fibers constitute about two-thirds of the fibers in the human body . . . Myelinated fibers are relatively large, with outer radii of 0.5 – 10 μm. They are wrapped with many layers of myelin between the nodes of Ranvier . . . In the myelinated region the conduction of the nerve impulse can be modeled by electrotonus because the conductance of the myelin sheath is independent of voltage. At each node a regenerative Hodgkin-Huxley-type (HH-type) conductance change restores the shape of the pulse. Such conduction is called saltatory conduction because saltare is the Latin verb “to jump.”

I have never liked the physical picture of an action potential jumping from one node to the next. The problem with this idea is that the action potential is distributed over many nodes simultaneously as it propagates along the axon. Consider an action potential with a rise time of about half a millisecond. Let the radius of the axon be 5 microns. Table 6.2 in Intermediate Physics for Medicine and Biology indicates that the speed of propagation for this axon is 85 m/s, which implies that the upstroke of the action potential is spread over (0.5 ms) × (85 mm/ms) = 42.5 mm. But the distance between nodes for this fiber (again, from Table 6.2) is 1.7 mm. Therefore, the action potential upstroke is distributed over 25 nodes! The action potential is not rising at one node and then jumping to the next, but it propagates in a nearly continuous way along the myelinated axon. I grant that in other cases, when the speed is slower or the rise time is briefer, you can observe behavior that begins to look saltatory (e.g., Huxley and Stampfli, Journal of Physiology, Volume 108, Pages 315–339, 1949), but even then the action potential upstroke is distributed over many nodes (see their Fig. 13).

If saltatory conduction is not the best description of propagation along a myelinated axon, then what is responsible for the speedup compared to unmyelinated axons? Primarily, the action potential propagates faster because of a reduction of the membrane capacitance. Along the myelinated section of the membrane, the capacitance is low because of the many layers of myelin (N capacitors C in series result in a total capacitance of C/N). At a node of Ranvier, the capacitance per unit area of the membrane is normal, but the area of the nodal membrane is small. Adding these two contributions together leads to a very small average, or effective, capacitance, which allows the membrane potential to increase very quickly, resulting in fast propagation.

In summary, I don’t find the idea of an action potential jumping from node to node to be the most useful image of propagation along a myelinated axon. Instead, I prefer to think of propagation as being nearly continuous, with the reduced effective capacitance increasing the speed. This isn’t the typical explanation found in physiology books, but I believe it’s closer to the truth. Rather than using the term saltatory conduction, I suggest we use curretory conduction, for the Latin verb currere, “to run.”