Adrien-Marie Legendre

On page 181 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce Legendre polynomials. The Legendre polynomial P₂(cos(θ)) arises naturally when calculating the extracellular potential in a volume conductor at a position far from an active nerve axon. We include the footnote “You can learn more about Legendre polynomials in texts on differential equations or, for example, in Harris and Stocker (1998).” On page 184, we list the first four Legendre polynomials (and have another footnote referring to Harris and Stocker). Any physics student should memorize at least the first three of these polynomials:

P₀(x) = 1

P₁(x) = x

P₂(x) = (3 x² – 1)/2 .

Legendre polynomials have many interesting properties. They are a solution of Legendre’s differential equation

(1–x²) d²P_n/dx² – 2 x dP_n/dx + n(n+1) P_n = 0 .

You can calculate any Legendre polynomial using Rodrigues formula

P_n(x) = 1/(2ⁿ n!) dⁿ((x²–1)ⁿ)/dxⁿ .

They form an orthogonal set of functions for x over the range from −1 to 1, which is rather too technical of a property to explain in this blog entry, but is very important.

The astute reader might note that Legendre’s differential equation is second order, so there should be two solutions. That is right, but the other solution—called a Legendre function of the second kind, Q_n—is rarely used, and tends to be poorly behaved at x = 1 and x = –1. For instance

Q₀(x) = ½ ln((1+x)/(1–x)) .

A definitive source for information about Legendre polynomials is the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun.

When do Legendre’s polynomials appear in physics? You often find them when working in spherical coordinates, especially when (to use an analogy with the earth) a function depends on latitude but not longitude (axisymmetry). For instance, the general axisymmetric solution to Laplace’s equation in spherical coordinates is a series of powers of the radius r multiplied by Legendre polynomials with x = cos(θ), where θ is measured from the z-axis (or, to use the earth analogy again, from the north pole). Take an introductory class in electricity and magnetism (from, say, the book by Griffiths), and you will use Legendre polynomials all the time.

Why do I bring up Legendre polynomials today? Regular readers of this blog may recall my recent obsession with all things French. Adrien-Marie Legendre (1752–1833) was a French mathematician. Details of his life are given in A Short Account of the History of Mathematics, by Rouse Ball.

Adrian Marie Legendre was born at Toulouse on September 18, 1752, and died at Paris on January 10, 1833. The leading events of his life are very simple and may be summed up briefly. He was educated at the Mazarin College in Paris, appointed professor at the military school in Paris in 1777, was a member of the Anglo-French commission of 1787 to connect Greenwich and Paris geodetically; served on several of the public commissions from 1792 to 1810; was made a professor at the Normal school in 1795; and subsequently held a few minor government appointments. The influence of Laplace was steadily exerted against his obtaining office or public recognition, and Legendre, who was a timid student, accepted the obscurity to which the hostility of his colleague condemned him.

Legendre’s analysis is of a high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original. His chief works are his Géométrie, his Théorie des nombres, his Exercices de calcul intégral, and his Fonctions elliptiques. These include the results of his various papers on these subjects. Besides these he wrote a treatise which gave the rule for the method of least squares, and two groups of memoirs, one on the theory of attractions, and the other on geodetical operations.

Augustin-Jean Fresnel

Apparently, dear reader, I am still obsessed by my trip to Paris last summer, because this will be the third week in a row that this blog has been about a famous French scientist. I hope you enjoy it.

Diffraction is a fundamental topic in physical optics that receives scant attention in the 4th edition of Intermediate Physics for Medicine and Biology. The index contains no entry for diffraction. (By the way, Russ Hobbie and I worked hard to make the index as complete and useful as possible.) However, a search for the term "diffraction" yields many appearances. Often it shows up as part of the term “x-ray diffraction,” but I have already addressed that technique in this blog a few weeks ago. A footnote on page 327, in Chapter 12 about images, mentions interference and diffraction in the context of coherence, and diffraction appears several times when discussing point-spread functions in that chapter. In Chapter 13 on ultrasound, diffraction is mentioned again as representing a limit to our ability to obtain an image. In Chapter 14, diffraction is discussed as a factor limiting our visual acuity.

The study of diffraction has a fascinating history, going back to the fundamental work of the French physicist Augustin-Jean Fresnel (1788-1827). Fresnel makes only one brief appearance in Intermediate Physics for Medicine and Biology, when discussing diffraction effects and the “Fresnel Zone” produced by an ultrasound transducer. To try and make up for Fresnel’s absence from our book, I will provide here some of the highlights of his short life (he died at age 39). Incidentally, I’m not the only blogger interested in Fresnel.

Waves and Grains:
Reflections on Light and Learning,
by Mark Silverman.

I first came to appreciate Fresnel’s contributions when reading the books of physicist Mark Silverman. In particular, I enjoyed Silverman’s Waves and Grains: Reflections on Light and Learning. He writes

Fresnel, as the reader will discover (if it is not already obvious), is a central figure and something of a hero in this book. Pathetically all too human in his desperate desire to distinguish himself in the world of science, his ambitions are the ambitions of all of us who do research, write papers, and seek recognition. As a young man trained in engineering, he first turned his attention to industrial chemistry but learned to his chagrin that what he thought was original work was anticipated by others. Disappointed, he later immersed himself in the wave theory of light, guided and encouraged by Francois Arago—one of very few wave enthusiasts in the Paris Academy—who helped publicize his work both in France and abroad…

In 1817 the Paris Academy launched a competition for the essay best accounting for the diffraction of light. With the exception of Arago, the committee responsible for the event consisted exclusively of partisans, like Laplace and Biot, of the particle hypothesis [of light…] Fresnel, as one might imagine, was not initially enthusiastic about entering—his whole direction of research having apparently already been ruled out by the wording. Nevertheless, urged on again by Arago, he composed a lengthy paper summarizing his philosophical approach, his methods, and his results. It is an amusing irony of history that Simeon-Denis Poisson—another graduate of the Polytechnique noted for his broad theoretical contributions to physics and mathematics, and a staunch advocate of the corpuscular theory—noted a glaring inconsistency in Fresnel’s theory. Applying this theory to an opaque circular screen, Poisson deduced the (to him) ludicrous result that the center of the shadow (doit) etre aussi eclaire que si l’ecran n’existait pas (must be as brightly illuminated as if the screen did not exist). Arago performed the experiment in advance of the committee’s decision, and the bright center—which history records as Poisson’s spot—showed up as predicted.

Fresnel, his relentless efforts finally recognized, received the prize—but Biot, Poisson, and other remained unshaken in their particle convictions.

If you get a copy of Silverman’s book, don’t miss the last chapters on Science and Learning.

Living here in Michigan, surrounded by the Great Lakes, I’ve become fond of lighthouses, and particularly with the spectacular Fresnel lenses that you can find in many of them. Click here to see pictures of some, and here to see information about Fresnel lenses found in Michigan. It is another of Fresnel’s many contributions to science.

Joseph Fourier

The August 2010 issue of Physics Today, published by the American Institute of Physics, contains an article by T. N. Narasimhan about “Thermal Conductivity Through the 19th Century.” A large part of the article deals with Joseph Fourier (17687–1830), the French physicist and mathematician. Russ Hobbie and I discuss Fourier’s mathematical technique of representing a periodic function as a sum of sines and cosines of different frequencies in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. Interestingly, this far-reaching mathematical idea grew out of Fourier’s study of heat conduction and thermal conductivity. Russ and I introduce thermal conductivity in Homework Problem 15 of Chapter 4 about diffusion. This is not as odd as it sounds because, as shown in the problem, heat conduction and diffusion are both governed by the same partial differential equation, typically called the diffusion equation (Eq. 4.24). The concept of heat conduction is crucial when developing the bioheat equation (Chapter 14), which has important medical applications in tissue heating and ablation.

Narasimhan’s article provides some interesting insights into Fourier and his times.

In 1802, upon his return to France from Napoleon’s Egyptian campaign, Fourier was appointed perfect of the department of Isere. Despite heavy administrative responsibilities, Fourier found time to study heat diffusion. He was inspired by deep curiosity about Earth and such phenomena as the attenuation of seasonal temperature variations in Earth’s subsurface, oceanic and atmospheric variations in Earth’s subsurface, oceanic and atmospheric circulation driven by solar heat, and the background temperature of deep space…

Thermal conductivity, appropriate for characterizing the internal conduction, was defined by Fourier as the quantity of heat per unit time passing through a unit cross-section divided by the temperature difference of two constant-temperature surfaces separated by unit distance… Fourier presented his ideas in an unpublished 1807 paper submitted to the Institut de France.

Fourier was not satisfied with the 1807 work. It took him an additional three years to go beyond the discrete finite-difference description of flow between constant-temperature surfaces and to express heat flow across an infinitesimally thin surface segment in terms of the temperature gradient.

When Fourier presented his mathematical theory, the nature of heat was unknown… Fourier considered mathematical laws governing the effects of heat to be independent of all hypotheses about the nature of heat… No method was available to measure flowing heat. Consequently, in order to demonstrate that his mathematical theory was physically credible, Fourier had to devise suitable experiments and methods to measure thermal conductivity.

It is not widely recognized that in his unpublished 1807 manuscript and in the prize essay he submitted to the Institut de France in 1811, Fourier provided results from transient and steady-state experiments and outlined methods to invert exponential data to estimate thermal conductivity. For some reason, he decided to restrict his 1822 masterpiece, The Analytical Theory of Heat, to mathematics and omit experimental results.

For more insight on Fourier’s life and times, see Keston’s article “Jospeh Fourier: Policitian and Scientist.” It begins

The life of Baron Jean Baptiste Joseph Fourier (1768–1830) the mathematical physicist has to be seen in the context of the French Revolution and its reverberations. One might say his career followed the peaks and troughs of the political wave. He was in turns: a teacher; a secret policeman; a political prisoner; governor of Egypt; prefect of Isère and Rhône; friend of Napoleon; and secretary of the Académie des Sciences. His major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur) changed the way scientists think about functions and successfully stated the equations governing heat transfer in solids. His life spanned the eruption and aftermath of the Revolution; Napoleon's rise to power, defeat and brief return (the so-called Hundred Days); and the Restoration of the Bourbon Kings.

Jean Leonard Marie Poiseuille, Biological Physicist

Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology contains an analysis of the flow of a viscous fluid through a pipe. Russ Hobbie and I show that the fluid flow is proportional to the fourth power of the pipe radius. We then state that

This relationship was determined experimentally in painstaking detail by a French physician, Jean Leonard Marie Poiseuille, in 1835. He wanted to understand the flow of blood through capillaries. His work and knowledge of blood circulation at that time have been described by Herrick (1942).

The paper by Herrick appeared in my favorite journal, the American Journal of Physics (J. F. Herrick, “Poiseuille’s Observations on Blood Flow Lead to a New Law in Hydrodynamics,” Volume 10, Pages 33–39, 1942). The key paragraph in the paper is quoted below.

The important role which the physical sciences have played in the progress of the biological sciences has eclipsed, more or less, the contributions which biologists have made to the physical sciences. Some of these contributions have become such an integral part of the physical sciences that their origin seems to have been forgotten. An outstanding example of such a contribution is that by Jean Leonard Marie Poiseuille (1799–1869). About 100 years ago Poiseuille brought a fundamental law to that division of physics known as hydrodynamics—which is a branch of rheology, according to more recent terminology. This law resulted indirectly from his observations on the capillary circulation of certain animals. Most physicists, chemists and mathematicians associate the name of Poiseuille with the phenomenon of viscosity because the cgs absolute unit for the viscosity coefficient has been named the poise in his honor. Few know the story leading up to the discovery of the law which bears his name. This law had more fundamental significance than Poiseuille himself realized. It established an excellent experimental method for the measurement of viscosity coefficients of liquids. The underlying principle of this method is in use today. Since Poiseuille’s law was based entirely on experiment, it was purely empirical. However, the law can be obtained theoretically. Those who are familiar with only the theoretical development are generally surprised to learn that the law was originally determined experimentally—and still more surprised to know that Poiseuille got his idea from studying the character of the flow of blood in the capillaries of certain animals.

More about Poiseuille and his law can be found in a paper by Pfitzner (“Poiseuille and His Law,” Anaesthesia, Volume 31, Pages 273–275, 1976)

Jean Leonard Marie Poiseuille (1791–1869) was born and died in Paris. Remarkably little seems to be known about his life. He studied medicine for a considerable time and submitted a thesis for his Doctorate in 1828 (aged 30–31 years). Where he carried out his early experiments studies, and how they were financed, is obscure.

His published work includes… “Experimental Studies on the Movement of Liquids in Tubes of Very Small Diameter” (his most famous paper, completed in 1842 and published in 1846). For his work “On the Causes of the Movement of the Blood in the Capillaries” he was awarded the Paris Academie des Sciences prize for experimental physiology. In later life he became a foundation member of the Academie de Medecine of Paris.

My biggest question about Poiseuille is the pronunciation of his name. I gather that it is pronounced pwah-zweez. The unit of the poiseuille has been proposed for a pascal second (or, newton second per square meter), but is not commonly used.

Don’t throw away the cane

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I quote the article “Don’t Throw Away the Cane” by Walter Blount (Journal of Bone and Joint Surgery, Volume 38, Pages 695–708, 1956).

The patient with a wise orthopedic surgeon walks with crutches for six months after a fracture of the neck of the femur. He uses a stick for a longer time—the wiser the doctor, the longer the time. If his medical adviser, his physical therapist, his friends, and his pride finally drive him to abandon the cane while he still needs one, he limps. He limps in a subconscious effort to reduce the strain on the weakened hip. If there is restricted motion, he cannot shift his body weight, but he hurries to remove the weight from the painful hip joint when his pride makes him reduce the limp to a minimum. The excessive force pressing on the aging hip takes its toll in producing degenerative changes. He should not have thrown away the stick.

I recently looked up the full article, which is delightful. It was listed as a JBJS classic in 2003 (Volume 85, Page 380). Here are a few more quotes. I suggest you read the entire paper.

As the causes of premature death are conquered one by one, man is given a longer life in which to grow old gracefully. In the twilight years that his forefathers rarely knew, he needs help in seeing, hearing, chewing, and walking. Gradually we are coming to look upon eye glasses, hearing devices, and dentures as welcome aids to gracious living rather than as the stigmata of senility. They should be accepted eagerly as components of a richer life. The cane, too, should be restored to favor as a means of preventing fatigue and a halting gait, rather than maligned as a sign of deterioration.

The use of the cane in order to prevent strain upon an ailing hip or knee is not generally accepted. In the patient's mind there is a nice distinction between the permissible use of a stick postoperatively and the adoption of this humble support for no other reason than the relief of a slight physical infirmity. A fat lady may waddle like a duck when she laboriously walks a few steps, but she resents the suggestion that she carry a cane. She would look much better with a stick than with the limp; and with support she could walk enough to get some exercise. More walking would help with weight reduction. But no! she is not ready for a cane yet! The patient with residual disability after poliomyelitis and with a fatiguing, unsightly lurch needs a cane. Early degenerative hip disease may require no treatment other than weight reduction and a stick in the opposite hand; however it takes an impressive orthopaedic surgeon to sell the idea…

As Pauwels has shown so well (Fig. 8), the use of a cane in the left hand reduces the pressure on the right femoral head without the need for limping. The support afforded by the stick greatly lessens the pull required of the abductor muscles in helping to support the body weight. The cane works through a long lever, so that a moderate push on the stick greatly relieves the strain on the hip [my boldface]. The relative forces are shown in Table I. Pauwels estimated that during the stance phase of walking, without the support of a cane, an average person exerts a static force of 385 pounds on the stationary hip. This weight can be reduced to 220 pounds by pushing down on a stick with the opposite hand the equivalent of 20 pounds. The cane is really an efficient mechanical device…

I should rather be remembered as a thoughtful surgeon than as a bold one. I submit that a well planned sequence of lesser operations with long intervals between, and the use of a cane as needed, may prove better for the patient and productive of a more desirable end result than some more heroic surgical procedure. There is a tendency among orthopaedic surgeons to exchange simple methods for dramatic treatment that will not require the use of the cane. The surgeon looks for a single, definitive, bridge-burning operation that will cure the patient completely for the rest of his life. Too often, this goal is not reached. The patient still needs the stick (or even crutches) after this heroic operation. If a satisfactory arthroplasty or reconstruction operation is performed, how much better it would be for most patients to urge the continued use of a cane in order to preserve the function of the reshaped bone by taking the strain off the hip for years, not for months only.

Blount was a leading physician and surgeon in orthopedics. His grandfather was a civil war surgeon, his mother was a physician and surgeon, and his sister was a pediatrician. He attended the University of Illinois and Rush Medical College. He helped develop the Milwaukee brace for spinal malalignment, was an expert on fractures in children, and introduced tibial stapling for epiphyses. In 1954 he became president of the American Academy of Orthopedic Surgeons.

The Anger Camera

In Section 17.12 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the gamma camera, used in to produce images during nuclear medicine procedures. The gamma camera often goes by another name, the Anger camera, after its inventor Hal Anger (1920–2005). Anger’s contributions to medical physics imaging rank him with giants of the field such as Godfrey Hounsfield, Allan Cormack, and Paul Lauterbur. Yet, it is harder to find information about Anger than about these other luminaries. His obituary in the New York Times stated

Mr. Anger was known in his field for inventing the gamma camera, which was first exhibited in 1958 at a meeting of the Society of Nuclear Medicine. The device, also known as a scintillation camera and later as the Anger camera, produced images of internal processes by tracking tiny amounts of radioactive substances, known as radiopharmaceuticals, given to patients.

The invention and later improvements represented a major advance in the diagnosis and treatment of brain tumors, bone marrow disorders and other life-threatening diseases.

Another obituary in the IEEE publication The Institute wrote

Seen by many as a quiet genius who shaped the future of nuclear medicine, Hal took a hands-on approach to science that also led to his invention of the well counter, which is used daily in nuclear medical labs worldwide to measure small quantities of radioactive substances. He also invented the whole-body scanner, the positron camera, and the multiplane tomographic scanner.

Nuclear medicine has been profoundly affected by Hal Anger. Millions of patients have benefited from diagnosis and treatment that depended on the Anger camera and the innovations made possible by its development.

Anger described his invention in a paper titled simply “Scintillation Camera” (Review of Scientific Instruments, Volume 29, Pages 27–33, 1958). The abstract is reproduced below.

A new and more sensitive gamma-ray camera for visualizing sources of radioactivity is described. It consists of a lead shield with a pinhole aperture, a scintillating crystal within the shield viewed by a bank of seven photomultiplier tubes, a signal matrix circuit, a pulse-height selector, and a cathode-ray oscilloscope. Scintillations that fall in a certain range of brightness, such as the photopeak scintillations from a gamma-ray-emitting isotope, are reproduced as point flashes of light on the cathode-ray tube screen in approximately the same relative positions as the original scintillations in the crystal. A time exposure of the screen is taken with an oscilloscope camera, during which time a gamma-ray image of the subject is formed from the flashes that occur. One of many medical and industrial uses is described, namely the visualization of the thyroid gland with I¹³¹.

The barn

Figure 15.2 of the 4th edition of Intermediate Physics for Medicine and Biology shows the cross section for the interaction of photons with carbon versus photon energy. The caption of the figure says “The cross section is given in barns: 1 b = 10⁻²⁸ m².” Where did this strange unit come from?

The July 1972 issue of Physics Today published a letter by M. G. Holloway and C. P Baker, explaining “How the Barn was Born.”

Some time in December of 1942, the authors, being hungry and deprived temporarily of domestic cooking, were eating dinner in the cafeteria of the Union Building of Purdue University. With cigarettes and coffee the conversation turned to the topic uppermost in their minds, namely cross sections. In the course of the conversation, it was lamented that there was no name for the unit of cross section of 10⁻²⁴ cm². It was natural to try to remedy this situation.

The tradition of naming a unit after some great man closely associated with the field ran into difficulties since no such person could be brought to mind. Failing in this, the names Oppenheimer and Bethe were tried, since these men had suggested and made possible the work on the problem with which the Purdue project was concerned. The “Oppenheimer” was discarded because of its length, although in retrospect an “Oppy” or “Oppie” would seem to be short enough. The “Bethe” was thought to lend itself to confusion because of the widespread use of the Greek letter. Since John Manley was directing the work at Purdue, his name was tried, but the “Manley” was through to be too long. The “John” was considered, but was discarded because of the use of the term for purposes other than as the name of a person. The rural background of one of the authors then led to the bridging of the gap between the “John” and the “barn.” This immediately seemed good and further it was pointed out that a cross section of 10⁻²⁴ cm² for nuclear processes was really as big as a barn. Such was the birth of the “barn.”

To the best knowledge of the authors, the first public (if it may be called that) use of the barn was in Report LAMS-2 (28 June, 1943) in which the barn was defined as a cross section of 1 × 10⁻²⁴ cm².

The authors would like to insist that the “barn” is spelled just that way, that no capital “b” is needed, and that the plural is “barns” with no letter “e” involved, and that the symbol be a small “b.” The meanings of “millibarn” and “kilobarn” are obvious.

Iron, Nature’s Universal Element

Iron, Nature's Universal Element:
Why People Need Iron
and Animals Make Magnets,
by Eugenie Mielczarek.

A few months ago in this blog, I mentioned that I put the book Iron, Nature’s Universal Element: Why People Need Iron and Animals Make Magnets, by Eugenie Mielczarek and Sharon McGrayne, on my list of things to read this summer. Well, I finished this book, and I recommend it. Russ Hobbie and I cite the book in Chapter 8 of the 4th edition of Intermediate Physics for Medicine and Biology.

Magnetism is used for orientation by several organisms. A history of studies in this area is provided in a very readable book by Mielczarek and McGrayne (2000).

My favorite part of Iron, Nature’s Universal Element was Chapter 4, on magnetotactic bacteria. Russ and I discussed these interesting little creatures in Section 8.8.3

Several species of bacteria contain linear strings of up to 20 particles of magnetite, each about 50 nm on a side encased in a mambrane [Frankel et al. (1979); Moskowitz (1995)]. Over a dozen different bacteria have been indentified that synthesize these intracellular, membrane-bound particles or magnetosomes (Fig. 8.25). Bacteria in the northern hemisphere have been shown to seek the north pole. Because of the tilt of the earth’s field, they burrow deeper into the environment in which they live. Similar bacteria in the southern hemisphere burrow down by seeking the south pole. In the laboratory the bacteria align themselves with the local field.

The caption to our Figure 8.25 reads

The small black dots are magnetosomes, small particles of magnetite in the magnetotactic bacterium Aquaspirillum magnetotacticum. The vertical bar is 1 [micron] long. The photograph was taken by Y. Gorby and was supplied by N. Blakemore and R. Blakemore, University of New Hampshire.

Mielczarek and McGrayne provide a colorful description of how magnetotactic bacteria were discovered.

“I see it crystal clearly,” Richard Blakemore said, recalling the evening he discovered Earth’s smallest living magnets. “I get excited every time I look at them.”

It was already dark outside the laboratory as Blakemore, peering through his microscope, searched through mud samples for bacteria. At twenty-three, Blakemore was a second-year graduate student in microbiology at the University of Massachusetts in Amherst. In 1975, fledgling microbiologists there were often assigned such simple tasks as identifying the material between their teeth or analyzing organisms in mud. His professor had collected the mud from a Massachusetts marsh, and asked Blakemore to learn everything possible about some large spiral bacteria in it. But that night, Blakemore said, “other organisms forced their existence on me...”

So while Blakemore looked through the scope, [advanced graduate student John Bresnick] picked up a magnetic stirrer lying beside the microscope and brought it up behind the swimmers. “Fortunately,” Blakemore recalled, “he had the end of the magnet pointing toward them so that it attracted them. And—all of a sudden—en masse—this whole massive population of bacteria swims in exactly the opposite way across the microscope stage. It was incredible, just incredible, and no one even believed my response. They thought I was kidding—until they looked in...” At that point, John Bresnick said, “I think you’ve discovered something...” “From then on,” Blakemore said, still starry-eyed more than twenty years later, “it was a night of incredulity...”

Blakemore’s microbiology professor was in Italy at the time, and Blakemore was exploding with the news, so he raced home to tell his wife Nancy. Abandoning all grammar in the joy of the memory, Blakemore said “It couldn’t have been perfecter. I didn’t really—hardly—know how to take it in.”

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.