Happy Birthday, Robert Resnick

Physics,
by Halliday and Resnick.

Robert Resnick, physics textbook author extraordinaire, was born 100 years ago last Wednesday (January 11, 1923). A Rensselaer Polytechnic Institute website states

Robert Resnick is professor emeritus at Rensselaer and the former Edward P. Hamilton Distinguished Professor of Science Education, 1974–93. Together with his co-author David Halliday, he revolutionized physics education with their now famous textbook on general physics, still one of the most highly regarded texts in the field today.  

He is author or co-author of seven physics textbooks, which appear in 15 editions and more than 47 languages. 

Resnick introduced Rensselaer’s interdisciplinary science curriculum in 1973 and was its chair for 15 years. He was awarded the American Association of Physics Teachers’ highest honor, the Oersted Medal, in 1975, and served as its president, 1986–90. A Distinguished Service Citation issued in 1967 by the association said, “Few physicists have had greater or more direct influence on undergraduate physics students than has Robert Resnick.” 

Rensselaer named its Robert Resnick Center for Physics Education in his honor. 

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

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite Resnick’s famous introductory physics book with Halliday once, but we cite his modern physics textbook—Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles—with Robert Eisberg many times. In fact, in IPMB we reproduce (with permission) five of Eisberg and Resnick’s figures. I remember studying from their textbook as an undergraduate physics major at the University of Kansas.

Resnick died nine years ago (January 29, 2014). To learn more about his remarkable life, you can read his obituary in Physics Today. 

Happy 100th birthday, Robert Resnick. We miss ya.

An excerpt from Robert Resnick’s Oersted Medal Lecture.

https://www.youtube.com/watch?v=THPGQDLdeHw&t=16s

Edith Anne Stoney, The First Woman Medical Physicist

Edith Anne Stoney

Today is Edith Anne Stoney’s birthday; she was born on January 6, 1869. In an article that appeared in the December, 2013 issue of Scope (the quarterly magazine of the Institute for Physics and Engineering in Medicine), Francis Duck describes Stoney as “the first woman medical physicist.” This week’s blog post includes excerpts from Duck’s fascinating article.

Edith Anne Stoney (1869–1938) was born in Dublin into a scientific family…. Her sister Florence became a radiologist and was awarded the OBE [Order of the British Empire].

Stoney began her education in math and physics, then later switched to medicine.

As a young woman, Edith demonstrated considerable mathematical talent, gaining a scholarship at Newnham College, Cambridge, where she achieved a First in the Part I Tripos examination in 1893. Extraordinarily, she was never awarded her Cambridge degree: women were excluded from graduation, a situation that would not change for another 50 years. She was later awarded [bachelor’s and master’s] degrees from Trinity College Dublin, after they accepted women in 1904. Career possibilities for university women were limited. She carried out some difficult calculations on gas turbines and searchlight design for Sir Charles Parsons, and then took a mathematics teaching post at Cheltenham Ladies’ College.

Stoney pioneered the role of physics in medical education, a task appreciated by readers of Intermediate Physics for Medicine and Biology.

The 1876 Medical Act had made it illegal for academic institutions to prevent access to medical education on the basis of gender. Anticipating this change in the law, the London School of Medicine for Women was established in 1874 as the first medical school for women in Britain. It soon became part of the University of London, with clinical teaching at the Royal Free Hospital. Edith’s sister Florence studied there, obtaining her [medical degree] in 1898. By this time, changed regulations had embedded physics firmly into medical training, and Edith gained an appointment as a physics lecturer there in 1899.

She became interested in medical imaging through her sister, the first female radiologist in the United Kingdom.

In 1901, the Royal Free Hospital appointed Florence into a new part-time position of medical electrician. The two sisters set about selecting, purchasing and installing x-ray equipment and, the following April, a new x-ray service was opened in the electrical department.

Edith and Florence with their
father George Johnstone Stoney.

In the early years of the 20th century, Stoney was part of the effort by women in Great Britain to gain the right to vote.

During the next few years Edith actively supported the women’s suffrage movement, though opposed the direct violent action with which it was later associated. The years from 1910–1915 did not go smoothly for her. After her father’s death in 1911 she no longer had his guidance to call on. As student numbers increased so did her staff, but they often did not stay long, finding her difficult to work with. Finally, in March 1915, she left [her teaching position at the University of London].

World War I began in 1914; Great Britain was fighting alongside France and Russia (and its ally Serbia) against Germany, Austria-Hungary, and Italy.

Edith was now free from other commitments and could make her own contribution to the war. She contacted the Scottish Women’s Hospitals (SWH), an organisation formed in 1914 to give medical support in the field of battle, financed by the women’s suffrage movement. In May she set off to Europe, and would be away for most of the next 4 years… She established stereoscopy to localise bullets and shrapnel and introduced the use of x-rays in the diagnosis of gas gangrene… [The war resulted in] traumatically injured soldiers and difficult working conditions. It could have crushed a weaker character… It was hard physical work for the women to pack up the whole tented hospital, weighing three or four hundred tons.

Stoney’s work with x-rays work put her at risk of radiation exposure.

In March 1918, and for the third time, she had to supervise a camp closure and retreat, when Villers-Cotterets was overrun by the advancing front. During the final months of the war the fighting intensified and there was a huge increase in workload. In the month of June 1918 alone the x-ray workload peaked at over 1,300, partly resulting from an increased use of fluoroscopy... However, [fluoroscopy] also resulted in an increased incidence of radiation burns to Edith’s staff, two of whom had to take sick leave to recover.

After the war ended, her work supporting the troops was honored by government awards, but not with an appropriate job.

Her war service was recognised by the medals that she was awarded: from France, the Médaille des épidémies and the Croix de Guerre; from Serbia, the Order of St Sava; and the Victory [Medal] and [the] British War Medal from Britain. Returning to England and with no pension and no medical qualification she took a post as lecturer in physics in the Household and Social Science department at King’s College for Women, which she held until 1925.

She retired in 1925, but remained active supporting women in science.

After leaving King’s she retired to Bournemouth where she lived with Florence who was by then terminally ill with spinal cancer. She supported the British Federation [of] University Women (BFUW) for which she had acted as the first treasurer before the war. She travelled widely, first with her ailing sister, and then alone after Florence died in 1932.

Stoney passed away just as Europe was hurtling toward another world war.

Edith Stoney died, aged 69 years, on 25th June 1938. Obituaries were printed in Nature, The Lancet and The Times…. She was not noted as a creative scientist: this was not her forte. She was a tough and single-minded woman with high academic ability. Her organisational skills established physics laboratories and courses in two institutes of higher education. She showed considerable bravery and resourcefulness in the face of extreme danger, and imagination in contributing to clinical care under the most difficult conditions of war. She was a strong advocate of education for women... At a time when medical physics was still struggling to become an identified profession, Edith Stoney stands out as one of its most able pioneers.

Anyone searching for a female role model in medical physics need look no further. What an amazing life. 

Happy birthday, Edith Anne Stoney. We need more like you today.

The Development of Transcranial Magnetic Stimulation

When I worked at the National Institutes of Health, I studied transcranial magnetic stimulation. In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe this technique to activate neurons in the brain.

Since a changing magnetic field generates an induced electric field, it is possible to stimulate nerve or muscle cells without using electrodes. The advantage is that for a given induced current deep within the brain, the currents in the scalp that are induced by the magnetic field are far less than the currents that would be required for electrical stimulation. Therefore transcranial magnetic stimulation (TMS) is relatively painless.

The method was invented in 1985 and when I arrived at NIH in 1988 the field was new and ripe for analysis. I spent the next seven years calculating electric fields in the brain and determining how the electric field couples to a nerve.

Roth, B. J. (2022) The Development of
Transcranial Magnetic Stimulation,
BOHR International Journal of
Neurology and Neuroscience,
Volume 1, Pages 8–20.

Recently, I wrote a review article telling the story of how transcranial magnetic stimulation began. You can get a copy at https://journals.bohrpub.com/index.php/bijnn/article/view/28; it is an open access article so everyone is free to download it. The abstract states

This review describes the development of transcranial magnetic stimulation in 1985 and the research related to this technique over the following 10 years. It not only focuses on work done at the National Institutes of Health but provides a survey of other related research as well. Key topics are the calculation of the electric field produced during magnetic stimulation, the interaction of this electric field with a long nerve axon, coil design, the time course of the magnetic stimulation pulse, and the safety of magnetic stimulation.

Readers of this blog will recognize some of the topics from earlier posts, such as the calculation of the induced electric field, determining the site of stimulation along a peripheral nerve, Paul Maccabee’s wonderful article, the four-leaf coil, the heating of metal electrodes, implantable microcoils, and Tony Barker's online interview. You could almost say I pre-wrote much of the review using this blog as my test bed. 

I like magnetic stimulation because it's a classic example of how a fundamental concept from physics can have a major impact in biology and medicine. If you combine this review of transcranial magnetic stimulation together with my earlier review of the bidomain model of cardiac tissue, you get a pretty good summary of my most important research.

Enjoy!

Think Before You Calculate!

I encourage students to build their qualitative problem solving skills by recasting equations in dimensionless variables, analyzing the limiting behavior of mathematical expressions, and sketching plots showing how functions behave. “Think Before You Calculate!” is my mantra. But how, specifically, do you do this? Let me show you an example.

Fig. 2.16 from IPMB. A plot of the solution
of the logistic equation when y₀ = 0.1,
y_∞ = 1.0, b₀ = 0.0667. Exponential
growth with the same values of
y₀ and b₀ is also shown.

In Section 2.10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the logistic model.

Sometimes a growing population will level off at some constant value. Other times the population will grow and then crash. One model that exhibits leveling off is the logistic model, described by the differential equation

dy/dt = b₀ y (1 – y/y_∞) ,                           (2.28)

where b₀ and y_∞ are constants….

If the initial value of y is y₀, the solution of Eq. 2.28 is

y(t) = 1 / [1/y_∞ + (1/y₀ – 1/y_∞) e^−b₀t] .    (2.29)

Below is a new homework problem, analyzing the logistic equation in a way to build insight. Consider it an early Christmas present. Santa won’t give you the answer, so you need to solve the problem yourself to gain anything from this post.

Section 2.10

Problem 36 ½. Consider the logistic model.

(a) Write Eq. 2.28 in terms of dimensionless variables Y and T, where Y = y/y_∞ and T = b₀t.

(b) Express the solution Eq. 2.29 in terms of Y, T, and Y₀ = y₀/y_∞.

(c) Verify that your solution in part (b) obeys the differential equation you derive in part (a).

(d) Verify that your solution in part (b) is equal to Y₀ at T = 0.

(e) In a plot of Y(T) versus T, which of the three constants (y_∞, y₀, and b₀) affect the qualitative shape of the solution, and which just scale the Y and T axes? 

(f) Verify that your solution in part (b) approaches 1 as T goes to infinity.

(g) Find an expression for the slope of the curve Y = Y(T). What is the slope at time T = 0? For what value of Y₀ is the initial slope largest? For what values of Y₀ is the slope small?

(h) The plot in Fig. 2.16 compares the solution of logistic equation with the exponential Y = Y₀ e^T. The figure gives the impression that the exponential is a good approximation to the logistic curve at small times. Do the two curves have the same value at T = 0? Do the two curves have the same slope at T = 0?

(i) Sketch plots of Y versus T for Y₀ = 0.0001, 0.001, 0.01, and 0.1.

(j) Rewrite the solution from part (b), Y = Y(T), using the constant T₀, where T₀ = ln[(1−Y₀)/Y₀]. Show that varying Y₀ is equivalent to shifting the solution along the T axis. What value of Y₀ corresponds to T₀ = 0?

(k) How does the logistic curve behave if Y₀ > 1? Sketch a plot of Y versus T for Y₀ =1.5.

(l) How does the logistic curve behave if Y₀ < 0? Sketch a plot of Y versus T for Y₀ = –0.5.

(m) Plot Y versus T for Y₀ = 0.1 on semilog graph paper.

If you solve this new homework problem and want to compare you solution to mine, email me at roth@oakland.edu and I’ll send you my solution. 

The Logistic Equation, MIT OpenCourseWare

https://www.youtube.com/watch?v=TCkLSYxx21c&t=69s

Mark Hallett Festschrift

Mark Hallett

Last Monday I attended (over the internet) a Festschrift to honor the retirement of Mark Hallett from the intramural program at the National Institutes of Health. Russ Hobbie and I cite Hallett in Chapter 8 of Intermediate Physics for Medicine and Biology.

Magnetic stimulation can be used to diagnose central nervous system diseases that slow the conduction velocity in motor nerves without changing the conduction velocity in sensory nerves (Hallett and Cohen 1989).

The reference is to a wonderful paper that Hallett and Leo Cohen wrote in the Journal of the American Medical Association.

Hallett M, Cohen LG (1989) Magnetism: A new method for stimulation of nerve and brain. JAMA 262:538–541.

Hallett came to NIH in 1984 and worked there for almost 40 years. I collaborated with him in the early 1990s, when I was working in NIH’s Biomedical Engineering and Instrumentation Program. My role was to calculate the electric field induced in the brain during transcranial magnetic simulation.

For a long time, Hallett was the clinical director for the National Institute of Neurological Disorders and Stroke intramural program. According to Google Scholar, his papers are cited about ten times every day, and his h-index is over 100, meaning he has published over 100 papers that have each been cited over 100 times. He has had an enormous impact on the field of neurophysiology. In particular, he trained an amazing number of young scholars who have gone on to be leaders in the field themselves, many of who spoke at the event.

Mark Hallett

Below is Hallett’s biography found on his NIH webpage.

Dr. Hallett obtained his A.B. and M.D. at Harvard University, had his internship in Medicine at the Peter Bent Brigham Hospital and his Neurology training at Massachusetts General Hospital. He had fellowships in neurophysiology at the NIH and in the Department of Neurology, Institute of Psychiatry in London, where he worked with C. David Marsden. Before coming to NIH in 1984, Dr. Hallett was the Chief of the Clinical Neurophysiology Laboratory at the Brigham and Women's Hospital in Boston and progressed to Associate Professor of Neurology at Harvard Medical School. He is currently Chief of the Medical Neurology Branch and Chief of its Human Motor Control Section. He is now Past-President of the International Federation of Clinical Neurophysiology. He has been President of the International Parkinson and Movement Disorder Society and Vice-President of the American Academy of Neurology. He served as Editor in Chief of Clinical Neurophysiology. Among many awards, in 2012 he became an Honorary Member of the American Neurological Association, and in 2014 won the Lifetime Achievement Award of the American Association of Neuromuscular and Electrodiagnostic Medicine. In 2017 he received the degree of Doctor of Medicine Honoris Causa from the University of Hamburg, and in 2018 was made an Honorary Member of the European Academy of Neurology. His research activities focus on the physiology of human voluntary movement and its pathophysiology in disordered voluntary movement and involuntary movement.

Once Hallett told me that he started out college studying physics, but when his instructor explained to his class that a magnetic field is just a relativistic effect of an electric field (see Problem 5 in Chapter 8 of IPMB) he switched to a premed program! 

At the end of his Festschrift, Hallett spoke and honored his many mentors. His final words were "I will be retiring, but not too much."

Enjoy your retirement, Mark Hallett, but not too much. Working with you was a delight.

https://www.youtube.com/watch?v=eR-D9bLWKhQ 

Oral History 2013: Stanley Fahn Interviews Mark Hallett

Surface Tension

Air and Water,
by Mark Denny.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I don’t talk much about surface tension. However, in Air and Water, Mark Denny devotes several chapters to it. He begins

We are now at the first of three chapters in which we explore the physics of the interface between water and air. As we will soon discover, the interface is a bizarre and fascinating place. To begin with, it is truly two-dimensional: it has neither outside nor inside. It is not contained in the water nor is it contained in the air; it is simply the place at which they meet. As such, its properties are not those of air or water alone, but of their mutual interaction, and these can be both surprising and nonintuitive…

We begin this exploration with an examination of the phenomenon known as surface tension. This is the force that keeps water droplets spherical, and it has a variety of biological consequences. For instance, we will see how surface tension allows trees to grow to majestic heights and flies to adhere to glass; how surface tension allows some insects to breathe under water and others to walk upon it.

Surface tension arises because a water molecule is usually hydrogen bonded to several other water molecules surrounding it. On the water surface, however, there is at least one hydrogen bond missing, so a higher surface energy is required to produce additional surface area. The surface energy of water in air is denoted γ and has a value of 0.07 J/m². This is a fairly high surface energy compared to most other liquids (for instance, everyone’s favorite, ethanol, has a surface energy of only 0.02 J/m²).

After explaining the origin of surface energy, Denny adds an important caveat.

In nature it is extremely rare to find an air-water interface that is not fouled to some extent with an ill-defined organic film. Most biological molecules (fatty acids in particular) can lower the surface energy to a fraction of that found in pure water. As a result, the surface of all but the cleanest bodies of water is likely to have a lower surface energy than reported here.

Denny then relates the concept of surface energy to that of surface tension.

To this point we have discussed the air-water interface in terms of its surface energy. Why, then, is this chapter entitled “Surface Tension”? It turns out that surface tension is just another way of expressing surface energy.

In the abstract, this is easily seen by comparing the units of the two expressions. Surface energy is expressed as J m⁻². But a joule is one newton meter, so J m⁻² is the same as N m⁻¹, that is, force per distance, a tension.

Surface tension is related to the concept of capillarity. Water tends to adhere to a clean glass surface. Therefore, water will rise in a hollow glass tube until it reaches a height at which the weight of the column is balanced by the adhesion force. This height is proportional to the surface tension and is inversely proportional to the radius of the tube. Denny observes that

Because the water in the tube is, in essence, hanging from the air-water interface, it is at a lower pressure than that of the surrounding air. As a result, if one were to poke a hole in the side of the tube, air would be drawn in rather than water being forced out.

Denny shows how capillarity is used to get water up a tree. Even more interesting is his discussion of insect tracheae.

Consider, for instance, the problems faced by insects. These animals rely on their tracheae and tracheoles to deliver oxygen to their muscles and to remove carbon dioxide… This system works only because the tracheae are tracheoles are filled with air. If these small tubes become filled with water, the rate at which they transport O₂ and CO₂ decreases 10,000-fold [because of the 100-fold difference of the diffusion constant in air and water]. How does the respiratory system of insects keep from filling up with water via capillarity?

The answer is likely to be that the inner surface of the respiratory system is coated with some substance that is not wetted by water… If the tracheae and tracheoles are coated with a waxy substance similar to that found on the external cuticle of many insects, water has no tendency to fill the system, and effective respiration is possible.

Denny then analyzes the law of Laplace, relating the air pressure inside a bubble to its radius and water’s surface tension. Russ and I analyze the case of a spherical bubble in Homework Problem 60 of Chapter 3 in IPMB. 

Denny concludes the chapter by examining animals that can walk on water.

The surface of lakes and streams provides a unique opportunity for terrestrial organisms. An animal that can walk on water has available to it a flat substratum from which to hunt aquatic prey and a refuge on which to escape from predators. There is just one problem: How does an animal manage to walk on water?

Surface tension, of course, provides the answer. If the animal contacts the water with a nonwettable structure of sufficient perimeter, the upward force of surface tension can support the organism’s weight. 

This only works for small animals.

The prospect of walking on water becomes less likely the larger the animal becomes. The crux of the problem is that an animal’s weight increases as the cube of a linear measure of its size, whereas the force due to surface tension increases in direct proportion to a linear measure (in this case, perimeter). For example, a mouse with a mass of 100 g weights 1 N. If the animal had feet coated with the same wax available to a water strider, it would need a perimeter of 40.3 m in contact with the water, roughly 10 m per foot. Feet of this size would clearly be impractical, and for this reason animals the size of mice do not walk on water.

The Neper

When discussing the attenuation of sound in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

In acoustics, the attenuation is usually expressed in decibels per meter.

At the bottom of the page is this footnote:

Sometimes the attenuation coefficient is expressed in nepers m⁻¹, in which case the natural logarithm of the intensity or pressure ratio is used.

The neper? What’s that?

First, let’s review the decibel. If the pressure amplitude of two sound waves are p₁ and p₂, their relative pressure can be written as

            20 log₁₀(p₂/p₁).

When expressed in this way, the pressure difference is said to be in decibels (dB). If p₂ is ten times p₁, then the pressure difference is 20 log₁₀(10) or 20 dB.

Often we express sound in terms of intensity rather than pressure. The intensity is proportional to the pressure squared, so the relative intensity difference of I₁ and I₂ is

            10 log₁₀(I₂/I₁).

The leading factor of 20 in the expression containing the pressures is replaced by 10 in the expression containing intensities. If you don’t like that factor of ten out front, you could not use it, in which case your intensity difference is expressed in the rarely used unit of bels rather than decibels.

Notice that the decibel is defined using of a logarithm with base ten, also known as the common logarithm. Alternatively, we could use the natural logarithm, with base e = 2.718..., which leads to the neper. However—and this is the confusing part—instead of having the leading factor of 20 in the expression for decibels in terms of pressure, the expression for nepers has no leading factor at all. A factor of 2 is removed because the neper is defined in terms of the pressure and not intensity, and a factor of 10 is removed because nepers are like bels and not decibels. So,

            ln(p₂/p₁)

is the pressure difference in nepers. If you insist on using intensity rather than pressure, you must use the ugly-looking expression

            ½ ln(I₂/I₁) .

A ten-fold difference in intensity is 1.15 nepers (Np), so 10 dB is the same as 1.15 Np, or 1 Np = 8.7 dB. If a sound wave attenuates at a rate of 1 neper per meter that means for every meter traveled the pressure falls by a factor of e and the intensity falls by a factor of 7.4. In tissue, attenuation is usually proportional to frequency, so as a rule of thumb the attenuation is about 100 dB per meter per megahertz or roughly 12 neper per meter per megahertz.

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

Where does the strange name “neper” come from? It honors the inventor logarithms, John Napier. Here is a excerpt about Napier from Asimov's Biographical Encyclopedia of Science & Technology.

NAPIER, John (nay’pee-ur) 

Scottish mathematician

Born: Merchiston Castle, near Edinburgh, 1550 

Died: Merchiston Castle, near Edinburgh, April 4, 1617

... Napier’s solid reputation rests upon a new method of calculation that first occurred to him in 1594… It occurred to Napier that all numbers could be expressed in exponential form. That is, 4 can be written as 2², while 8 can be written as 2³, and 5, 6, and 7 can be written as 2 to some fractional power between 2 and 3. Once numbers were written in such exponential form, multiplication could be carried out by adding exponents, and division by subtracting exponents. Multiplication and division would at once become no more complicated than addition and subtraction.

Napier spent twenty years working out rather complicated formulas for obtaining exponential expressions for various numbers. He was particularly interested in the exponential forms of the trigonometric functions, for these were used in astronomical calculations and it was these which Napier wanted to simplify. His process of computing the exponential expressions led him to call them logarithms (“proportionate numbers”) and that is the word still used.

Finally, in 1614, Napier published his tables of logarithms, which were not improved on for a century, and they were seized on with avidity. Their impact on the science of the day was something like that of computers on the science of our own time. Logarithms then, like the computers now, simplified routine calculations to an amazing extent and relieved working scientists of a large part of the noncreative mental drudgery to which they were subjected.

Reduced Mass

In Section 14.4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss molecular energy levels. In particular, we examine translational, rotational, and vibrational levels. When analyzing rotational levels, we consider a simple diatomic molecule and divide the motion into two parts: a uniform translation of the center of mass, and a rotation about the center of mass. We show that the rotational energy can be written as ½Iω², where ω is the rotational angular frequency and I is the moment of inertia. The moment of inertia is I = [m₁ m₂/(m₁ + m₂)] R², where m₁ and m₂ are the mass of the two atoms making up the diatomic molecule, and R is the distance between them. In quantum mechanics, the spacing of rotational energy levels depends on I.

Later in the same section, Russ and I consider vibrational motion. However, we don’t do a detailed analysis for a diatomic molecule, like we did for rotational motion. In this blog post, I will remedy that situation and present the analysis of vibrations of a diatomic molecule. Our goal is to derive an expression for the vibration frequency in terms of the masses of the two atoms and the spring constant connecting them.

Let’s do the analysis in one dimension. Consider two atoms with mass m₁ and m₂ connected by a spring with spring constant k. The position of m₁ is x₁, and the position of m₂ is x₂.

First, write down Newton’s second law for each atom.

    m₁ d²x₁/dt²  = − k (x₁ − x₂ ) ,

    m₂ d²x₂/dt²  =    k (x₁ − x₂ ) .

Next, define two new variables, as we did for rotational motion: x, the position of the center of mass, and X, the distance between the two masses

     x = [m₁/(m₁+m₂)] x₁ + [m₂/(m₁+m₂)] x₂ ,

    X = x₁ − x₂ .

Then, rewrite Newton’s second law in terms of x and X. After some algebra, we get two equations

     (m₁+m₂) d²x/dt²  =  0 

     [m₁m₂/(m₁+m₂)] d²X/dt²  =   − kX

The first equation represents a free particle of mass M, where

     M = m₁+m₂ ,

and the second equation represents a bound particle with spring constant k and mass m (often called the reduced mass)

     m = m₁m₂/(m₁+m₂) . 

The angular frequency of the vibration is therefore

     ω = √(k/m)

(If you don’t follow that last step, see Appendix F of IPMB).

We have reached our goal: the angular frequency of the vibration, ω, written in terms of k, m₁, and m₂

     ω = √[k (m₁+m₂)/m₁m₂]

In quantum mechanics, the energy levels depend on ω, and therefore on the reduced mass m.

If m₁ >> m₂ then m is approximately m₂. Likewise, if m₂ >> m₁ then m is approximately m₁. For example, if you want the vibration frequency of hydrogen chloride (HCl), the reduced mass is close to the mass of the hydrogen atom.

If m₁ = m₂ = μ (like for molecules such as O₂ and N₂), then the reduced mass m is equal to μ/2. It’s that factor of two in the denominator that’s the surprise.

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

The relationship between m, m₁, and m₂ can be written

    1/m = 1/m₁ + 1/m₂ .

This looks just like the equation for adding resisters in parallel. 

If you want to learn more, I suggest looking at Chapter 12 of Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Robert Eisberg and Robert Resnick, often cited in IPMB.

Randy Travis

Forever and Ever, Amen,
by Randy Travis.

I’m a big fan of country music. After all, I was a graduate student in Music City: Nashville. I used to ride my bike down to 16th Avenue by the original Country Music Hall of Fame and listen to the up-and-coming singers perform on the street. During the late 1980s, just as I was finishing my dissertation, the biggest country star was Randy Travis. His debut album, Storms of Life, appeared in 1986, and for the next several years he dominated the country music scene.

I recently listened to Travis’s 2019 autobiography, Forever and Ever, Amen. It tells the story of his glory years, but also covers his troubled youth, his time as the singing cook at the Nashville Palace nightclub, and his tragic health problems.

In 2013 Travis was incapacitated by a massive stroke. The most common type of stroke occurs when a clot blocks the flow of blood to part of the brain. Stroke is ranked as the fifth leading cause of death in the United States; every four minutes someone dies of a stroke. Many of those that survive have brain damage. Following his stroke, Travis suffered from limited use of his right hand and severe speech impairment.

The question for readers of Intermediate Physics for Medicine and Biology is, how can physics address stroke? Two applications that are important for stroke diagnosis and treatment are Diffusion Tensor Imaging and Transcranial Magnetic Stimulation. In diffusion tensor imaging, diffusion in the brain is measured using strong gradient magnetic fields applied during magnetic resonance imaging. Diffusion is anisotropic in the brain’s white matter, with water diffusing faster parallel to nerve axon tracts than perpendicular to them. In IPMB, Russ Hobbie and I write

Diffusion is usually greater along the direction of the nerve or muscle fibers. Since the orientation of the fibers changes throughout the body, the elements of the diffusion tensor vary as well. However, some features of the diffusion tensor, such as the trace (see Prob. 49), are independent of the fiber direction, and are particularly useful when monitoring diffusion in anisotropic tissue, such as the white matter of the brain. In addition, the diffusion tensor contains information about the fiber direction, allowing one to map fiber tract trajectories noninvasively using MRI (Basser et al. 2000).

Diffusion can serve as a biomarker to diagnose stroke and to monitor recovery.

Transcranial magnetic stimulation (TMS) is a method to excite neurons in the brain. Russ and I describe it as

Magnetic stimulation can be used to diagnose central nervous system diseases that slow the conduction velocity in motor nerves without changing the conduction velocity in sensory nerves (Hallett and Cohen 1989). It could be used to monitor motor nerves during spinal cord surgery, and to map motor brain function. Because TMS is noninvasive and nearly painless, it can be used to study learning and plasticity (changes in brain organization over time; Wassermannet al. 2008). Recently, researchers have suggested that repetitive TMS might be useful for treating disorders such as depression (O’Reardon et al. 2007) and Alzheimer’s disease (Freitas et al. 2011).

You could add stroke to the list of disorders that might benefit from repetitive transcranial magnetic stimulation. I say “might” because the technique is still being studied as a stroke therapy, but any method that influences brain plasticity has at least the potential to be useful to stroke victims.

Now, almost ten years after his stroke, Travis continues to slowly recover. Although he has not yet been able to return to a singing career, in 2016 he did lead his fans in singing Amazing Grace when he was inducted into the Country Music Hall of Fame. His autobiography is captivating and inspiring. The courage and tenacity of stroke victims should motivate us all to use our science to address this devastating illness.

Randy Travis sings Amazing Grace at his induction into the Country Music Hall of Fame.

https://www.youtube.com/watch?v=11bgiJH1zhA

Randy Travis singing his signature song, Forever and Ever, Amen.

https://www.youtube.com/watch?v=KtKXc_v2iLE

Randy Travis singing Storms of Life.

https://www.youtube.com/watch?v=piTt6zu2FKs

The Intellectual Immigration That Has Mattered Most to Biology

The Eighth Day of Creation,
by Horace Freeland Judson.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I analyze the role physics plays in the biological sciences. What is that role, and how did it begin? Insight can be found in Horace Freeland Judson’s classic book The Eighth Day of Creation: The Makers of the Revolution in Biology. The development of molecular biology occurred in the mid-twentieth century and was spurred in part by immigrant physicists escaping central Europe before the start of World War II. Judson describes this intellectual exodus from physics to biology.

The mass intellectual emigration from continental Europe in the 1930s, which so stimulated physical science in the United States and England, also had profound consequences for biology, even though the men involved were fewer and younger, with their reputations still to make. They included [Max] Perutz, who left Austria for England in 1936, and [Erwin] Chargaff, also an Austrian, who emigrated to the United States in 1934. A less direct influence was the distinguished and passionately intelligent Hungarian physicist Leo Szilard, who in the 1930s had been the first to envision the possibility that sustained nuclear fission, a chain reaction, would work and cause an explosion, and the first to urge that the United States should try to make an atomic bomb. Szilard wrote the letter about the idea that [Albert] Einstein signed and that was read to [President Franklin] Roosevelt in 1939. Szilard worked on the atomic project at the University of Chicago through the war; afterwards, in reaction against the weapons and against the big-money, big-team physics he had been instrumental in creating, he turned to biology and also to campaigning within the international scientific community for disarmament—for example, through the Pugwash conferences, which he helped to found. In 1947, [University of Chicago chancellor Robert] Hutchins gave Szilard the physicist an appointment as professor of biology and sociology. In the early years of molecular biology, Szilard was an erratic if interesting experimenter and theorist, a cross-pollinator of ideas and an effective critic of others’ work, an intellectual and ethical inspiration to younger scientists. 

The most important immigrant to biology, however, was Max Delbrück. Delbrück was German, born to the aristocracy of the intellect—his father was the professor of history and his uncle the professor of theology in the University of Berlin—and trained as a quantum physicist. His mind and style had been formed by Niels Bohr, the physicist, philosopher, poet, and incessant Socratic questioner who made Copenhagen one of the capital cities of science between the wars. Delbrück_’s ideas about the physical properties of the gene, in a youthful paper of 1935, had led [Austrian physicist Erwin] Schrödinger to write [the influential book] What Is Life? Delbrück was perhaps the earliest of the theoretical physicists who have crossed over to biology; Szilard, [Francis] Crick, Maurice Wilkins were others, while Linus Pauling, arriving at biology from a different tangent, was a physical chemist whose strength was founded in quantum mechanics. The move from physics has been the intellectual immigration that has mattered most to biology [my italics].

Each of us who has emigrated from physics to biology has followed in the footsteps of giants such as Szilard and Delbrück. We each follow our own individual path, but share a common bond. Physicists have played key roles in biology, and will continue to do so.