X-ray Crystallography

Two weeks ago in this blog, when reviewing Judson’s excellent book The Eighth Day of Creation, I wrote that X-ray crystallography played a central role in the development of molecular biology. But Russ Hobbie and I do not discuss X-ray crystallography in the 4th edition of Intermediate Physics for Medicine and Biology, even though it is a classic example of physics applied in the biomedical sciences. Why? I think one of the reasons for this is that Russ and I made the conscious decision to avoid molecular biophysics. In our preface we write

Biophysics is a very broad subject. Nearly every branch of physics has something to contribute, and the boundaries between physics and engineering are blurred. Each chapter could be much longer; we have attempted to provide the essential physical tools. Molecular biophysics has been almost completely ignored: excellent texts already exist, and this is not our area of expertise. This book has become long enough.

Nevertheless, sometimes—to amuse myself—I play a little game. I say to myself “Brad, suppose someone pointed a gun to your head and demanded that you MUST include X-ray crystallography in the next edition of Intermediate Physics for Medicine and Biology. Where would you put it?”

My first inclination would be to choose Chapters 15 and 16, about how X-rays interact with tissue and their use in medicine, which seems a natural place because crystallography involves X-rays. Yet, these two chapters deal mainly with the particle properties of X-rays, whereas crystallography arises from their wave properties. Also, Chapters 15 and 16 make a coherent, self-contained story about X-rays in medical physics for imaging and therapy, and a digression on crystallography would be out of place. An alternative is Chapter 14 about Atoms and Light. This is a better choice, but the chapter is already long, and it does not discuss electromagnetic waves with wavelengths shorter than those in the ultraviolet part of the spectrum. Chapter 12 on Images is another possibility, as crystallography uses X-rays to produce an image at the molecular level based on a complicated mathematical algorithm, much like tomography uses X-rays to predict an image at the level of the whole body. Nevertheless, if that frightening gun were held to my head, I believe I would put the section on X-ray crystallography in Chapter 11, which discusses Fourier analysis. It would look something like this:

11.6 ½ X-ray Crystallography

One application of the Fourier series and power spectrum is in X-ray crystallography, where the goal is to determine the structure of a molecule. The method begins by forming a crystal of the molecule, with the crystal lattice providing the periodicity required for the Fourier series. DNA and some proteins form nice crystals, and their structures were determined decades ago.* Other proteins, such as those that are incorporated into the cell membrane, are harder to crystallize, and have been studied only more recently, if at all (for instance, see the discussion of the potassium ion channel in Sec. 9.7).

X-rays have a short wavelength (on the order of Angstroms), but not short enough to form an image of a molecule directly, like one would obtain using a light microscope to image a cell. Instead, the image is formed by diffraction. X-rays are an electromagnetic wave consisting of oscillating electric and magnetic fields (see Chapter 14). When an X-ray beam is incident on a crystal, some of these oscillations add in phase, and the resulting constructive interference produces high amplitude X-rays that are emitted (diffracted) in some discrete directions but not others. This diffraction pattern (sometimes called the structure factor, F) depends on the wavelength of the X-ray and the direction (see Prob. 19 2/3). One useful result from electromagnetic theory is that the structure factor is related to the Fourier series of the electron density of the molecule: F is just the a_n and b_n coefficients introduced in the previous three sections, extended to account for three dimensions. Therefore, the electron density (and thus the molecular structure) can be determined if the structure factor is known.

A fundamental limitation of X-ray crystallography is that the crystallographer does not measure F, but instead detects the intensity |F|². To understand this better, recall that the Fourier series consists of a sum of both cosines (the a_n coefficients) and sines (b_n). You can always write the sum of a sine and cosine having the same frequency as a single sine with an amplitude c_n and phase d_n (See Prob. 19 1/3)

a_n cos(ω_n t) + b_n sin(ω_n t) = c_n sin(ω_n t + d_n) . (1)

The measured intensity is then c_n². In other words, an X-ray crystallography experiment allows you to determine c_n, but not d_n. Put in still another way, the experiment measures the power spectrum only, not the phase. Yet, in order to do the Fourier reconstruction, phase information is required. How to obtain this information is known as the “phase problem,” and is at the heart of crystallographic methods. One way to solve the phase problem is to measure the diffraction pattern with and without a heavy atom (such as mercury) attached to the molecule: some phase information can be obtained from the difference of the two patterns (Campbell and Dwek (1984)). In order for this method to work, the molecule must have the same shape with and without the attached heavy atom present.

* for a fascinating history of these developments, see Judson (1979)

Problem 19 1/3 Use the trigonometric identity sin(A+B) = sinA cosB + cosA sinB to relate a_n and b_n in Eq. (1) to c_n and d_n.

Problem 19 2/3 Bragg’s law can be found by assuming that the incident X-rays (having wavelength λ) reflect off a plane formed by the regular array of points in the lattice. Assume that two adjacent planes are separated by a distance d, and that the incident X-ray bean falls on this plane at an angle θ with respect to the surface. The condition for constructive interference is that the path difference between reflections from the two planes is an integral multiple of λ. Derive Bragg’s law relating θ, λ and d.

Campbell, I. D., and R. A. Dwek (1984) Biological Spectroscopy. Menlo Park, CA, Benjamin/Cummings.

Judson, H. F. (1979) The Eighth Day of Creation. Touchstone Books

For more information on X-ray crystallography, see http://www.ruppweb.org/Xray/101index.html or http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html.

AAPT Summer Meeting in Portland Oregon

On Tuesday, Russ Hobbie gave a talk about “Medical Physics in the Introductory Physics Course” at the American Association of Physics Teachers Summer Meeting in Portland Oregon. His session, with over 100 people attending, focused on Reforming the Introductory Physics Courses for Life Science Majors, a topic currently of great interest and one that I have discussed before in this blog. You can find the slides that accompanied his talk at the 4th edition of Intermediate Physics for Medicine and Biology website. His talk focused on five topics that he feels are crucial for the introductory course: 1) Exponential growth and decay, 2) Diffusion and solute transport, 3) Intracellular potentials and currents, 4) Action potentials and the electrocardiogram, and 5) Fitting exponentials and power laws to data. All these topics are covered in our book. Russ and I also compiled a list of topics for the premed physics course, and cross listed them to our book, this blog, and other sources. You can find the list on the book website, or download it here.

Our book website is a source of other important information. For instance, you can download the errata, containing a list of known errors in the 4th edition of Intermediate Physics for Medicine and Biology. You will find Russ’s American Journal of Physics paper “Physics Useful to a Medical Student” (Volume 43, Pages 121–132, 1975), and Russ and my American Journal of Physics “Resource Letter MP-2: Medical Physics” (Volume 77, Pages 967–978, 2009). Other valuable items include MacDose, a computer program Russ developed to illustrate the interaction of radiation with matter, a link to a movie Russ filmed to demonstrate concepts related to the attenuation and absorption of x rays, sections from earlier editions of Intermediate Physics for Medicine and Biology that were not included in the 4th edition, and a link to the American Physical Society, Division of Biological Physics December 2006 Newsletter containing an interview with Russ upon the publication of the 4th edition of our book. You can even find a link to the Intermediate Physics for Medicine and Biology facebook group.

Russ and I hope that all this information on the book website, plus this blog, helps the reader of Intermediate Physics for Medicine and Biology keep up-to-date, and increases the usefulness of our book. If you have other suggestions about how we can make our website even more useful, please let us know. Of course, we thank all our dear readers for using our book.

The Eighth Day of Creation

The Eighth Day of Creation:
The Makers of the Revolution in Biology,
by Horace Freeland Judson.

I recently finished reading The Eighth Day of Creation, a wonderful history of molecular biology by Horace Freeland Judson. The book is divided into three parts: 1) DNA—Function and Structure: the elucidation of the structure of deoxyribonucleic acid, the genetic material, 2) RNA—The Functions of the Structure: the breaking of the genetic code, the discovery of the messenger, and 3) Protein—Structure and Function: the solution of how protein molecules work. The first part centers on the story of how Watson and Crick discovered the double-helix structure of DNA, a story also told in Watson’s book The Double Helix (required reading for any would-be scientist). I was less familiar with the RNA tale in the second part, but was fascinated by the “Good Friday” meeting in which the various roles of RNA (as both a messenger taking the genetic information from DNA to the protein, and as part of ribosomes where protein synthesis takes place) was first understood by Sydney Brenner and Francois Jacob, among others. I was somewhat familiar with Kornberg’s deciphering of the genetic code from my days at the National Institutes of Health, where Kornberg worked. The last section tells how Max Perutz used X-ray crystallography to determine the structure of hemoglobin, the first protein structure known.

New to me was the story of Jacques Monod and his study of bacteria, which led to our understanding of how protein synthesis is controlled. Last week in this blog I mentioned seeing a display about Monod at the Pasteur museum in Paris. Particularly fascinating was the story of Monod’s role as a leader of the French resistance against the Nazis during World War II, and how he continued his scientific research while participating in the resistance. Judson writes

“In the autumn of 1943, a meeting was called in Geneva of representatives of all the armed groups of the French resistance, to coordinate their military actions. Just before the meeting, Philippe Monod heard from his brother that he was the delegate of the Francs-Tireurs from Paris. In November, the Gestapo arrested a minor agent of one of the main resistance networks in France, Reseau Velites, centered on the Ecole Normale Superieur. Marchal’s identity [Marchal was an alias used by Monod] and activities were known to the agent. Monod had to go underground completely, leaving his apartment, never sleeping more than a night or two at one address, staying away from the Sorbonne. On 14 February 1944, the Gestapo caught Raymond Croland, chief of the Reseau Velites, who knew Monod.

On the run from his own laboratory, Monod was given bench space by [Andre] Lwoff. “I don’t think I was ever searched for, actually,” he said. “But the possibility existed because at least one—in fact, several men had been picked up who knew what I was doing and who knew my name and where I worked. But it was known that I lived near the Sorbonne and worked at the Sorbonne, so the Gestapo would have had no reason to hunt for me at the Pasteur Institute.” In Lwoff’s laboratory, in collaboration with Alice Audureau, a graduate student, Monod that winter began a new set of experiments…

I would rank The Eighth Day of Creation second in my list of the best scientific histories I have read, just behind Richard Rhodes’ The Making of the Atomic Bomb, and just ahead of Bruce Hunt’s The Maxwellians. Interestingly, some of the characters who appeared in The Eighth Day of Creation also played a role in The Making of the Atomic Bomb: in particular, George Gamow and Leo Szilard (Szilard was mentioned in the very first sentence of Rhodes’ book). Readers of the 4th edition of Intermediate Physics for Medicine and Biology will be interested in learning that many of the pioneers in molecular biology were trained as physicists. Judson writes “new people came into biology, and most famously the physicists: Max Delbruck, Leo Szilard, Francis Crick, Maurice Wilkins, [and] on an eccentric orbit George Gamow.” I couldn’t help but be struck by the central role of X-ray crystallography in the history of molecular biology. Under physicist William Bragg’s leadership at the Cavendish, four Nobel prizes were awarded (in the same year, 1962) for molecular structure determination: Watson and Crick for DNA, and Perutz and Kendrew for the structure of hemoglobin and myoglobin. I highly recommend the book, especially for young biology students interested in the history of their subject.

I will end with the opening paragraphs of The Eighth Day of Creation, where Judson draws parallels between the revolutions in physics in the first decades of the 20th century and the revolution in biology in the middle of the century.

The sciences in our century, to be sure, have been marked almost wherever one looks by momentous discoveries, by extraordinary people, by upheavals of understanding—by a dynamism that deserves to be called permanent revolution. Twice, especially, since 1900, scientists and their ideas have generated a transformation so broad and so deep that it touches everyone’s most intimate sense of the nature of things. The first of these transformations was in physics, the second in biology. Between the two, we are most of us spontaneously more interested in the science of life; yet until now it is the history of the transformation of physics that has been told.

The revolution in physics came earlier. It began with quantum theory and the theory of relativity, with Max Planck and Albert Einstein, at the very opening of the century; it encompassed the interior of the atom and the structure of space and time; it ran through the settling of the modern form of quantum mechanics by about 1930. Most of what has happened in physics since then, at least until recently, has been the playing out of the great discoveries—and of the great underlying shift of view—of those three decades. The decades, that shift of view, the discoveries, and the men who made them are familiar presences, at least in the background, to most of us; after all, they built the form of the world as we now take it to be. The autobiographies of the major participants, their memoirs and philosophical reflections, have been composed, their biographies written in multiple—and they remain long in print, for these were men of intelligence, originality, and, often, eccentricity. The scientific papers have been scrutinized as historical and literary objects. The letters have been catalogued and published. The collaborations have been disentangled, the conferences reconvened on paper with vivid imaginative sympathy, the encounters, the conversations, even the accidents reconstructed.

The revolution in biology stands in contrast. Beginning in the mid thirties, its first phase, called molecular biology, came to a kind of conclusion—not an end, but a pause to regroup—by about 1970. A coherent if preliminary outline of the nature of life was put together in those decades. This science appeals to us very differently from physics. It directly informs our understanding of ourselves. Its mysteries once seemed dangerous and forbidden; its consequences promise to be practical, personal, urgent. At the same time, biology has been growing accessible to the general reader as it never was before and as the modern physics never can be. Indeed, part of the plausibility of molecular biology to the scientists themselves is that it is superbly easy to visualize. The nonspecialist can understand this science, at least in outline, as it really is—as the scientist imagines it. Yet the decades of these discoveries have hardly been touched by historians before now. The Eighth Day of Creation is a historical account of the chief discoveries of molecular biology, of how they came to be made, and of their makers—for these, also, though only two or three are yet widely known, were scientists, often of intelligence, originality, even eccentricity.

Paris

I just returned from a vacation in Paris, where my wife and I celebrated our 25th wedding anniversary. Russ Hobbie was there at the same time, although conflicting schedules did not allow us to get together. My daughter Katherine posted the blog entries for the last two weeks, when I had limited computer access. Thanks, Kathy.

Although most of our time was spent doing the usual tourist activities (for example, the Arc de Triomphe, the Notre Dame Cathedral, Versailles, and, my favorite, a dinner cruise down the Seine), I did keep my eye open for those aspects of France that might be of interest to readers of the 4th edition of Intermediate Physics for Medicine and Biology. We visited the Pantheon, where we saw the tomb of Marie Curie (a unit of nuclear decay activity, the curie, was named after her and is discussed on page 489 of Intermediate Physics for Medicine and Biology). Marie Curie lies next to her husband Pierre Curie (of the Curie temperature, page 216). Also in the Pantheon is Jean Perrin, who determined Avogadro’s number (see the footnote on page 85) and Paul Langevin, of the Langevin equation (page 87). Hanging from the top of the dome is a Foucault pendulum, in the exact place where Leon Foucault publicly demonstrated the rotation of the earth in 1851. I like it when physics takes center stage like that.

Another scientific site we visited is a museum honoring Louis Pasteur at the Pasteur Institute. Pasteur chose to be buried in his home (now the museum) rather than in the Pantheon. Readers of Intermediate Physics for Medicine and Biology will find him to be an excellent example of a researcher who bridges the physical and biological sciences. His first job was as a professor of physics, although he would probably be considered more of a chemist that a physicist. His early work was on chiral molecules and how they rotated light. He later became famous for his research on the spontaneous generation of life and a vaccine for rabies. In his book Adding A Dimension, Isaac Asimov lists Pasteur as one of the ten greatest scientists of all time. The museum is enjoyable, although it is not as accessible to English speakers as some of the larger museums such as the Louvre and the delightful Musee d’Orsay. Because I speak no French, I had a difficult time following many of the Pasteur exhibits. Also at the museum was a nice display about microbiologist Jacques Monod, who I will discuss in a future entry to this blog.

A Short History of Chemistry,
by Isaac Asimov.

The only other French scientist on Asimov’s top-ten list was the chemist Antoine Lavoisier. Oddly, the French don’t seem to celebrate Lavoisier’s accomplishments as much as you might expect. (Beware, my conclusion is based on a brief 2-week vacation, and I may have missed something.) Perhaps his death by the guillotine during the French Revolution has something to do with it. We visited the Place de la Concorde, where Lavoisier was beheaded. In A Short History of Chemistry, Asimov writes

In 1794, then, this man [Lavoisier], one of the greatest chemists who ever lived, was needlessly and uselessly killed in the prime of life. “It required only a moment to sever that head, and perhaps a century will not be sufficient to produce another like it,” said Joseph Lagrange, the great mathematician. Lavoisier is universally remembered today as “the father of modern chemistry.”

I normally associate Leonardo da Vinci with Italy, but when touring the Chateau at Amboise in the Loire Valley, we stumbled unexpectedly upon his grave. He spent the last three years of his life in France. We toured an excellent museum dedicated to da Vinci, containing life-size reconstructions of some of his engineering inventions. Although da Vinci had many interests and may be best known for his paintings (yes, I saw the Mona Lisa while at the Louvre), at least some of his work might be called biomedical engineering, such as his work on an underwater breathing apparatus and on human flight.

Seventy-two famous French scientists and mathematicians are listed on the Eifel Tower, including Laplace (of the Laplacian, page 91), Ampere (of Ampere’s law, page 206, and the unit of current, page 145), Navier (of the Navier-Stokes equation, page 27), Legendre (of Legendre polynomials, page 184), Becquerel (of the unit of activity, page 489), Fresnel (of the Fresnel zone for diffraction, page 352), Coulomb (of the unit of charge and Coulomb’s law, both on page 137), Poisson (of Poisson’s ratio, page 27; the Poisson-Boltzmann equation, page 230; and the Poisson probability distribution, page 572), Clapeyron (of the Clausius-Clapeyron relation, page 78), and Fourier (of the Fourier series, page 290). I could not see all these names because the tower was partially covered for painting. Note that Lavoisier was included on the Eiffel Tower, but Poiseuille (of Poiseuille flow, page 17) was not. The view from the top of the tower is spectacular.

I admit, I am not the best of travelers and am glad to be home in Michigan. But I believe there is much in France that readers of Intermediate Physics for Medicine and Biology will find interesting.

Reynolds Number

The Reynolds number is a key concept for anyone interested in biofluid dynamics. Russ Hobbie and I discuss the Reynolds number in Section 1.18 (Turbulant Flow and the Reynolds Number) of the 4th edition of Intermediate Physics for Medicine and Biology.

The importance of turbulence (nonlarminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by

N_R = L V ρ/η

where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs…

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V, and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces… The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive.”

Steven Vogel, in his fascinating book Life in Moving Fluids, describes the importance of the Reynolds number more elegantly.

The peculiarly powerful Reynolds number [is] the center piece of biological (and even nonbiological) fluid mechanics. The utility of the Reynolds number extends far beyond mere problems of drag; it’s the nearest thing we have to a completely general guide to what’s likely to happen when solid and fluid move with respect to each other. For a biologist, dealing with systems that span an enormous size range, the Reynolds number is the central scaling parameter that makes order of a diverse set of physical phenomena. It plays a role comparable to that of the surface-to-volume ratio in physiology.

The Reynolds number is named after the British engineer Osborne Reynolds (1842–1912). He developed the Reynolds number as a simple way to understand the transition from laminar to turbulent flow of fluids in a pipe. Perhaps it is fitting to let Reynolds have the last word. Below he describes experiments in which he added a filament of dye to the fluid (as quoted by Vogel in Life in Moving Fluids).

When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line across the tube. If the water in the tank had not quite settled to rest, as sufficiently low velocities, the streak would shift about the tube, but there was no appearance of sinuosity. As the velocity was increased by small stages, at some point in the tube, always at a considerable distance from the trumpet or intake, the colour band would all at once mix up with the surrounding water. Any increase in the velocity caused the point of break-down to approach the trumpet, but with no velocities that were tried did it reach this. On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls showing eddies.

Adolf Fick

Russ Hobbie and I discuss Fick’s laws of diffusion in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology. The German scientist Adolf Fick (1829–1901) was a classic example of a researcher who was comfortable in both physics and physiology. He enrolled at the University of Marburg with the goal of studying mathematics and physics, but eventually switched to medicine, and earned an MD in 1852. Of particular interest to me is that he wrote a classic textbook titled Medical Physics (1856), which was one of the first books on this topic. I have not read this book, which almost certainly is written in German (although I am half German through my father’s side, I cannot speak or read the language). Nevertheless, I wonder if Intermediate Physics for Medicine and Biology might be a descendant of this text.

Fick was only 26 when he proposed his two laws of diffusion. The first law (Eq. 4.18a in our book)—similar to Ohm’s law for electrical current or Fourier’s law for heat conduction—states that the diffusive flux is proportional to the concentration gradient. The constant of proportionality is the diffusion constant, which Fick first introduced. Fick’s second law (Eq. 4.24) arises by combining his first law with the equation of continuity (Eq. 4.2) and is what we generally refer to as the diffusion equation. He tested his two laws by measuring the diffusion of salt in water. He even noticed the strong temperature dependence of the diffusion constant.

Fick contributed to physiology and medicine in several ways. He made the first successful contact lens, and he developed a method to measure cardiac output based on oxygen consumption and blood oxygen concentration. You can find more information about his life at http://www.corrosion-doctors.org/Biographies/FickBio.htm.

Myopia

Section 14.12 in the 4th edition of Intermediate Physics for Medicine and Biology discusses the physics of the eye. One topic related to vision that I have always found fascinating is myopia.

In nearsightedness or myopia, parallel rays come to a focus in front of the retina. The eye is slightly too long for the shape of the cornea… The total converging power of the eye is too great, and the relaxed eye focuses at some closer distance, from which the rays are diverging. Accommodation can only increase the converging power of the eye, not decrease it, so the unassisted myopic eye cannot focus on distant objects. Myopia can be corrected by placing a diverging spectacle or contact lens in front of the eye, so that incoming parallel rays are diverging when the strike the cornea.

The interesting thing about myopia is that, in contrast to far-sightedness (hypermetropy), you cannot correct it by accommodation. Before the invention of eye glasses in the late Middle Ages, if you were born with myopia then distant objects would always be a blur.

Mornings on Horseback,
by David McCullough.

When teaching Biological Physics (PHY 325) at Oakland University, I often end my discussion of myopia with a quote from David McCullough’s wonderful biography of Theodore Roosevelt, Mornings on Horseback. Roosevelt suffered from myopia and didn’t get his first glasses until he was a teenager. McCullough tells the story:

Then, at a stroke, the summer of 1872, he was given a gun and a large pair of spectacles and nothing had prepared him for the shock, for the infinite difference in his entire perception of the world about him or his place in it.

The gun was a gift from Papa—a 12-gauge, double-barreled French-made (Lefaucheux) shotgun with a lot of kick and of such simple, rugged design that it could be hammered open with a brick if need be, the ideal gun for an awkward, frequently absent-minded thirteen-year-old. But blasting away with it in the woods near Dobbs Ferry he found he had trouble hitting anything. More puzzling, his friends were constantly shooting at things he never even saw. This and the fact that they could also read words on billboards that he could barely see, he reported to his father, and it was thus, at summer’s end, that the spectacles were obtained.

They transformed everything. They “literally opened an entirely new world to me,” he wrote years afterward, remembering the moment. His range of vision until then had been about thirty feet, perhaps less. Everything beyond was a blur. Yet neither he nor the family had sensed how handicapped he was. “I had no idea how beautiful the world was… I could not see, and yet was wholly ignorant that I was not seeing.”

How such a condition could possibly have gone undetected for so long is something of a mystery, but once discovered it did much to explain his awkwardness and the characteristic detached look he had, those large blue eyes “not looking at anything present.”

I am a lover of history, and a big fan of David McCullough. A couple of his books with a scientific or engineering bent are Path Between the Seas: The Creations of the Panama Canal and The Great Bridge: The Epic Story of the Building of the Brooklyn Bridge. His purely historical books, such as 1776 and John Adams, are also excellent.

To learn more, see the information about myopia on the website for the American Optometric Association. An modern option for correcting myopia that was not available in Roosevelt’s time is laser surgery to reshape the cornea.

The Gibbs Paradox

Last week in this blog, I wrote that the “Gibbs Paradox” deserved an entire entry of its own. Well, here it is. Russ Hobbie and I mention the Gibbs Paradox in a footnote in Section 3.18 (The Chemical Potential of a Solution) in the 4th edition of Intermediate Physics for Medicine and Biology. When calculating the entropy of mixing (where a solute and solvent are intermixed), we derived an expression for the number of ways N particles can be distributed among N sites. If we assume the solute particles are indistinguishable, there is only one way. The footnote then reads

The fact that there is only one mircostate because of the indistinguishability of the particles is called the Gibbs paradox. For an illuminating discussion of the Gibbs paradox, see Casper and Freier (1973).

Fundamentals of Statistical
and Thermal Physics,
by Frederick Reif.

The Gibbs Paradox is examined in more detail by Frederick Reif in his landmark textbook Fundamentals of Statistical and Thermal Physics. (Indeed, Chapter 3 of Intermediate Physics for Medicine and Biology follows a statistical approach similar to Reif’s analysis, and even more similar to the discussion in his introductory textbook—a personal favorite of mine—Statistical Physics, Berkeley Physics Course Volume 5). Reif considers “a gas consisting of N identical monatomic molecules of mass m enclosed in a container of volume V.” When he calculates the entropy, S, of the gas, he obtains

S = N k [ln V + 3/2 ln T + σ]     (7.2.16)

where k is Boltzmann’s constant, T is the absolute temperature, and σ is a constant independent of N, T, and V. He then ends the section with the provocative statement “This expression for the entropy is, however, not correct,” which leads to his discussion (Sec. 7.3) of the Gibbs paradox. Reif continues

The challenging statement at the end of the last section suggests that the expression (7.2.16) for the entropy merits some discussion… [The expression] for S is clearly wrong since it implies that the entropy does not behave properly as an extensive quantity. Quite generally, one must require that all thermodynamic relations remain valid if the size of the whole system under consideration is simply increased by a scale factor α, i.e., if all its extensive parameters are multiplied by the same factor α. In our case, if the independent extensive parameters V and N are multiplied by α, the mean energy… is indeed properly increased by this same factor, but the entropy S in (7.2.16) is not increased by α because of the term N ln V.

Indeed, (7.2.16) asserts that the entropy S of a fixed volume V of gas is simply proportional to the number N of molecules. But this dependence on N is not correct, as can readily be seen in the following way. Imagine that a partition is introduced which divides the container into two parts. This is a reversible process which does not affect the distribution of systems over accessible states. Thus, the total entropy ought to be the same with, or without, the partition in place; i.e.

S = S' + S"     (7.3.1)

where S' and S" are the entropies of the two parts. But the expression (7.2.16) does not yield the simple additivity required by (7.3.1). This is easily verified. Suppose, for example, that the partition divides the gas into two equal parts, each containing N' molecules of gas in a volume V'. Then the entropy of each part is given by (7.2.16) as

S' = S" = N' k [ln V' + 3/2 ln T + σ]

while the entropy of the whole gas without partition is by (7.2.16)

S = 2 N' k [ ln (2 V') + 3/2 ln T + σ] .

Hence

S – 2 S' = 2 N' k ln(2 V') – 2 N' k ln V' = 2 N' k ln2

and is not equal to zero as required by (7.3.1).

This paradox was first discussed by Gibbs and is commonly referred to as the “Gibbs paradox.” Something is obviously wrong in our discussion; the question is what.

Reif then analyzes in more detail the implications of removing the partition between the two sides of the box. He finds that

The act of removing the partition has thus very definite physical consequences. Whereas before removal of the partition a molecule of each subsystem could only be found within a volume V', after the partition is removed it can be located anywhere within the volume V = 2 V'. If the two subsystems consisted of different gasses, the act of removing the partition would lead to diffusion of the molecules throughout the whole volume 2V' and consequent random mixing of the different molecules. This is clearly an irreversible process; simply replacing the partition would not unmix the gases. In this case the increase in entropy in (7.3.2) would make sense as being simply a measure of the irreversible increase of disorder resulting from the mixing of unlike gases [the entropy of mixing that Russ and I calculated].

But if the gases in the subsystems are identical, such an increase of entropy does not make physical sense. The root of the difficulty embodied in the Gibbs paradox is that we treated the gas molecules as individually distinguishable, as though interchanging the positions of two like molecules would lead to a physically distinct state of the gas. This is not so. Indeed, if we treated the gas by quantum mechanics (as we shall do in Chapter 9), the molecules would, as a matter of principle, have to be regarded as completely indistinguishable. A calculation of the partition function would then automatically yield the correct result, and the Gibbs paradox would never arise. Our mistake has been to take the classical point of view too seriously. Even though one may be in a temperature and density range where the motion of molecules can be treated to a very good approximation by classical mechanics, one cannot go so far as to disregard the essential indistinguishability of the molecules.

In a sidenote, Reif adds

Just how different must molecules be before they should be considered distinguishable?… In a classical view of nature two molecules could, or course, differ by infinitesimal amounts… In a quantum description this troublesome question does not arise because of the quantized discreteness of nature… Hence the distinction between identical and nonidentical molecules is completely unambiguous in a quantum-mechanical description. The Gibbs paradox thus foreshadowed already in the last [19th] century conceptual difficulties that were resolved satisfactorily only by the advent of quantum mechanics.

Several good American Journal of Physics articles discuss the Gibbs phenomenon. Pesic examines Gibb’s own writings to trace his thoughts on the issue (“The Principle of Identicality and the Foundations of Quantum Theory: I. The Gibbs Paradox,” American Journal of Physics, Volume 59, Pages 971–974, 1991), and Landsberg and Tranah study in more detail in role of the Gibbs paradox for quantum mechanics (“The Gibbs Paradox and Quantum Gases,” American Journal of Physics, Volume 46, Pages 228–230, 1978). Finally, Casper and Freier (the authors of the paper cited in our footnote) analyze the Gibbs paradox by comparing macroscopic and microscopic points of view (“‘Gibbs Paradox’ Paradox,” American Journal of Physics, Volume 41, Pages 509–511, 1973).

You know, there is a lot of physics in that little footnote on page 68 of Intermediate Physics for Medicine and Biology.

The Gibbs Phenomenon

In chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Fourier analysis, a fascinating but very mathematical subject. One of the most surprising results of Fourier analysis is the Gibbs phenomenon, which we describe at the end of Sec. 11.5 (Fourier Series for a Periodic Function).

Table 11.4 shows the first few coefficients for the Fourier series representing the square wave, obtained from Eq. 11.34… Figure 11.16 shows the fits for n = 3 and n = 39. As the number of terms in the fit is increased, the value of Q [measuring the least squares fit between the function at its Fourier series] decreases. However, spikes of constant height (about 18% of the amplitude of the square wave or 9% of the discontinuity in y) remain. These are seen in Fig. 11.16. These spikes appear whenever there is a discontinuity in y and are called the Gibbs phenomenon.

You have to be amazed by the Gibbs phenomenon. Think about it: as you add terms in the sum, the fit between the function and its Fourier series gets better and better, but the overshoot in amplitude does not get any smaller. Instead, the region containing ringing near the discontinuity gets narrower and narrower. If you want to see a figure like our Fig. 11.16 presented as a neat animation, take a look at http://www.sosmath.com/fourier/fourier3/gibbs.html. Also, check out http://ocw.mit.edu/ans7870/18/18.06/javademo/Gibbs/ for an interactive demo that will let you include up to 200 terms in the Fourier series.

The Gibbs phenomenon is important in medical imaging. The entry for the Gibbs phenomenon from the Encyclopedia of Medical Imaging is reproduced below.

Gibbs phenomenon, (J. Willard Gibbs, 1839-1903, American physicist), phenomenon occurring whenever a “curve” with sharp edges is subject to Fourier analysis. The Gibbs phenomenon is relevant in MR imaging, where it is responsible for so-called Gibbs artefacts. Consider a signal intensity profile across the skull, where at the edge of the brain the signal intensity changes from virtually zero to a finite value. In MR imaging the measurement process is a breakdown of such intensity profiles into their Fourier harmonics. i.e. sine and cosine functions. Representation of the profiles measured with a limited number of Fourier harmonics is imperfect, resulting in high frequency oscillations at the edges, and the image can therefore exhibit some noticeable signal intensity variations at intensity boundaries: the Gibbs phenomenon, overshoot artefacts, or “ringing.” The artefacts can be suppressed by filtering the images. However, filtering can in turn reduce spatial resolution.

Figures 12.24 and 12.25 of our book show a CT scan with ringing inside the skull and its removal by filtering, an example of the Gibbs phenomenon.

Josiah Willard Gibbs was a leading American physicist from the 19th century. He is particularly well known for his contributions to thermodynamics. Gibbs appears at several places in Intermediate Physics for Medicine and Biology. Section 3.17 discusses the Gibbs free energy, a quantity that provides a simple way to keep track of the changes in total entropy when a system is in contact with a reservoir at constant temperature and pressure. A footnote on page 68 addresses the Gibbs paradox (which deserves an entire blog entry of its own), and Problem 47 in Chapter 3 introduces the Gibbs factor (similar to the Boltzmann factor but including the chemical potential).

Selected Papers of Great
American Physicists.

The preface to Gibbs’ book on statistical mechanics is reproduced in Selected Papers of Great American Physicists: The Bicentennial Commemorative Volume of the American Physical Society 1976, edited by Spencer Weart. I recall being quite impressed by this book when in graduate school at Vanderbilt University. Below is a quote from Weart’s biographical notes about Gibbs.

Gibbs, son of a Yale professor of sacred literature, descended from a long line of New England college graduates. He studied at Yale, received his Ph.D. there in 1863—one of the first doctorates granted in the United States—tutored Latin and natural philosophy there, and then left for three decisive years in Europe. Up to that time, Gibbs had shown interest in both mathematics and engineering, which he combined in his dissertation “On the Form of the Teeth in Wheels in Spur Gearing.” The lectures he attended in Paris, Berlin and Heidelberg, given by some of the greatest men of the day, changed him once and for all. In 1871, two years after his return from Europe, he became Yale’s first Professor of Mathematical Physics. He had not yet published any papers on this subject. For nine years he held the position without pay, living on the comfortable inheritance his father had left; only when Johns Hopkins University offered Gibbs a post did Yale give him a small salary.

Gibbs never married. He lived out a calm and uneventful life in the house where he grew up, which he shared with his sisters. He was a gentle and considerate man, well-liked by those who knew him, but he tended to avoid society and was little known even in New Haven. Nor was he known to more than a few of the world’s scientists—partly because his writings were extremely compact, abstract and difficult. As one of Gibb’s European colleagues wrote, “Having once condensed a truth into a concise and very general formula, he would not think of churning out the endless succession of specific cases that were implied by the general proposition; his intelligence, like his character, was of a retiring disposition.” The Europeans paid for their failure to read Gibbs: A large part of the work they did in thermodynamics before the turn of the century could have been found already in his published work.

Happy Birthday Laser!

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

This month marks the 50^th anniversary of the invention of the laser. In May 1960, Theodore Maiman built the first device to produce coherent light by the mechanism of “Light Amplification by Stimulated Emission of Radiation” at Hughes Research Laboratories in Malibu, making the laser just slightly older than I am. A special website, called laserfest, is commemorating this landmark event. Eisberg and Resnick discuss lasers in Section 11.7 of their textbook Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (quoted from the first edition, 1974).

In the solid state laser that operates with a ruby crystal, some Al atoms in the Al₂O₃ molecules are replaced by Cr atoms. These “impurity” chromium atoms account for the laser action… The level of energy E1 is the ground state and the level of energy E3 is the unstable upper state with a short lifetime (≈10⁻⁸ sec), the energy difference E3-E1 corresponding to a wavelength of about 5500 Å. Level E2 is an intermediate excited state which is metastable, its lifetime against spontaneous decay being about 3 x 10⁻³ sec. If the chromium atoms are in thermal equilibrium, the population number of the states are such that [n3 is less than n2 is less than n1]. By pumping in radiation of wavelength 5500 Å, however, we stimulate absorption of incoming photons by Cr atoms in the ground state, thereby raising the population number of energy state E3 and depleting energy state E1 of occupants. Spontaneous emission, bringing atoms from state 3 to state 2, then enhances the occupancy of state 2, which is relatively long-lived. The result of this optical pumping is to decrease n1 and increase n2, such that n2 is greater than n1 and population inversion exists. Now, when an atom does make a transition from state 2 to state 1, the emitted photon of wavelength 6943 Å will stimulate further transitions. Stimulated emission will dominate stimulated absorption (because n2 is greater than n1) and the output of photons of wavelength 6943 Å is much enhanced. We obtain an intensified coherent monochromatic beam.

Lasers are an important tool in biology and medicine. Russ Hobbie and I discuss their applications in Chapter 14 (Atoms and Light) the 4th edition of Intermediate Physics for Medicine and Biology. In Section 14.5 (The Diffusion Approximation to Photon Transport) we write

A technique made possible by ultrashort light pulses from a laser is time-dependent diffusion. It allows determination of both μ_s and μ_a [the scattering and absorption attenuation coefficients]. A very short (150-ps) pulse of light strikes a small region on the surface of the tissue. A detector placed on the surface about 4 cm away records the multiply-scattered photons… A related technique is to apply a continuous laser beam, the amplitude for which is modulated at various frequencies between 50 and 800 MHz. The Fourier transform of Eq. 14.29 gives the change in amplitude and phase of the detected signal. Their variation with frequency can also be used to determine μ_a and μ_s.

We also mention lasers in Section 14.10 (Heating Tissue with Light).

Sometimes tissue is irradiated in order to heat it; in other cases tissue heating is an undesired side effect of irradiation. In either case, we need to understand how the temperature changes result from the irradiation. Examples of intentional heating are hyperthermia (heating of tissue as a part of cancer therapy) or laser surgery (tissue ablation). Tissue is ablated when sufficient energy is deposited to vaporize the tissue.

Russ and I give many references about lasers in medicine in our Resource Letter (“Resource Letter MP-2: Medical Physics,” American Journal of Physics, Volume 77, Pages 967–978, 2009):

F. Lasers and optics

Lasers have introduced many medical applications of light, from infrared to the visible spectrum to ultraviolet.

150. Lasers in Medicine, edited by R. W. Waynant (CRC, Boca Raton, 2002). (I)

151. Laser-Tissue Interactions: Fundamentals and Applications, M. H. Niemz (Springer, Berlin, 2007). (I)

152. “Lasers in medicine,” Q. Peng, A. Juzeniene, J. Chen, L. O. Svaasand, T. Warloe, K.-E. Giercksky, and J. Moan, Rep. Prog. Phys. 71, Article 056701, 28 pages
(2008). (A)

A fascinating and fast-growing new technique to image biological tissue is optical coherence tomography “OCT.” It uses reflections like ultrasound but detects the reflected rays using interferometry.

153. Optical Coherence Tomography, M. E. Brezinski (Elsevier, Amsterdam, 2006). Overview of the physics of OCT and applications to cardiovascular medicine, musculoskeletal disease, and oncology. (I)

154. “Optical coherence tomography: Principles and applications,” A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, Rep. Prog. Phys. 66, 239–303 (2003). (I)

With infrared light, scattering dominates over absorption. In this case, light diffuses through the tissue. Optical imaging in turbid media is difficult but not impossible.

155. “Recent advances in diffuse optical imaging,” A. P. Gibson, J. C. Hebden, and S. R. Arridge, Phys. Med. Biol. 50, R1–R43 (2005). (I)

156. “Pulse oximetry,” R. C. N. McMorrow and M. G. Mythen, Current Opinion in Critical Care 12, 269–271 (2006). The pulse oximeter measures the oxygenation of blood and is based on the diffusion of infrared light. (I)

One impetus for medical applications of light has been the development of new light sources, such as free-electron lasers and synchrotrons. In both cases, the light frequency is tunable over a wide range.

157. “Free-electron-laser-based biophysical and biomedical instrumentation,” G. S. Edwards, R. H. Austin, F. E. Carroll, M. L. Copeland, M. E. Couprie, W. E. Gabella, R. F. Haglund, B. A. Hooper, M. S. Hutson, E. D. Jansen, K. M. Joos, D. P. Kiehart, I. Lindau, J. Miao, H. S. Pratisto, J. H. Shen, Y. Tokutake, A. F. G. van der Meer, and A. Xie, Rev. Sci. Instrum. 74, 3207–3245 (2003). (I)

158. “Medical applications of synchrotron radiation,” P. Suortti and W. Thomlinson, Phys. Med. Biol. 48, R1– R35 (2003). (I)

Finally, photodynamic therapy uses light-activated drugs to treat diseases.

159. “The physics, biophysics and technology of photodynamic therapy,” B. C. Wilson and M. S. Patterson, Phys. Med. Biol. 53, R61–R109 (2008). (A)

Happy birthday, laser!