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.  

The International Day of Medical Physics, Held on Marie Curie's Birthday

The poster for the 2022
International Day of Medical Physics

Monday, November 7, is the International Day of Medical Physics. The purpose of this annual event, organized by the International Organization for Medical Physics, is to raise awareness about the role that medical physics plays in our lives. The date coincides with the birthday of Marie Curie.

The Search for the Elements,
by Isaac Asimov.

To celebrate the International Day of Medical Physics, I quote an excerpt from Isaac Asimov’s book The Search for the Elements that describes Curie’s discovery of the elements polonium and radium.

Thomson, Roentgen, Becquerel, and Rutherford all received Nobel prizes for their work. But the most glamorous of all the Nobel laureates of the turn of the century was Marie Curie, born Marja Sklodowska in Poland in 1867. Marie went to Paris to get an education (at the Sorbonne), and there she met and married a French chemist, Pierre Curie.

Becquerel’s discovery of the radiations from uranium fascinated Marie; it was she who suggested the term “radioactivity.” With enthusiasm and imagination, she plunged into a career of investigating this phenomenon. Marie began by trying to measure the strength of radioactivity. As the instrument of measurement she used the phenomenon of piezoelectricity, involving the electrical behaviors of crystals, which had been discovered by Pierre Curie. Pierre, realizing perhaps that his wife was a greater scientist than he was, abandoned his own research and joined her.

As they measured the radioactivity of samples of uranium ore, they found to their surprise that some samples were many times more radioactive than could be accounted for by the uranium content. This could only mean that other radioactive elements also were present. But if so, the amount must be extremely small, because the Curies were unable to detect any by ordinary chemical analysis. So they decided they would have to collect huge quantities of the ore to get enough of the trace material to analyze. They managed to get tons of ore from the mines in Bohemia; the Austrian government had no use for it and was glad to give it away, provided the Curies paid for the transportation. This took almost all their life savings.

They set up shop in a little unheated shed and went to work on their mounds and mounds of uranium ore. Year after year they kept concentrating the radioactivity, discarding inactive material and working with the active. (Marie took time out to have a baby, Irene, who later turned out to be a great scientist on her own.) At last, in July 1898, they succeeded in boiling down their tons of ore to a highly radioactive residue. What they had was a pinch of black powder which was 400 times more radioactive than the same quantity of pure uranium would have been. In this bit of stuff they found a new element resembling tellurium. Mendeleev might have named it “eka-tellurium.” The Curies called it “polonium,” after Marie’s native land.

This element didn’t account for all the radioactivity, however. A still more active element must be hiding in their ore. Six months later they finally concentrated that element. Its properties were like those of barium. The element fitted into row IIa in the seventh period of Mendeleev’s table. It was the first new element discovered in the seventh period since Berzelius had found thorium 60 years before.

The Curies called the new element “radium,” because of its powerful radioactivity.

Pierre Curie died in 1906 as the result of a traffic accident (involving a horse-drawn cab, not one of the new-fangled motor cars). Marie took over his professorship at the Sorbonne and carried on alone. She was the first woman professor in the history of that proud institution. Moreover, she was the only scientist in history to receive two Nobel prizes—one in physics (shared with her husband and Becquerel) for their accurate measurements of radioactivity, and one in chemistry for the discovery of polonium and radium.

International Organization of Medical Physics President Message on the International Day of Medical Physics 2022. 

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

Marie Curie: Scientist

https://www.youtube.com/watch?v=ZEV4KJBJvEg