Stable Nuclei

Fig. 17.2 in Intermediate
Physics for Medicine and Biology.

In Figure 17.2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I show a plot of all the stable nuclei. The vertical axis is the number of protons, Z (the atomic number), and the horizontal axis is the number of neutrons, N (the neutron number). The mass number A equals Z + N. Each tiny black box in the figure corresponds to a stable isotope. 

This figure summarizes a tremendous amount of information about nuclear physics. Unfortunately, the drawing is too small to show much detail. We must magnify part of the drawing to tell what boxes correspond to what isotopes. In this post, I provide several such magnified views.

Figure 17.2, with part of the drawing magnified.

Let’s begin by magnifying the bottom left corner, corresponding to the lightest nuclei. The most abundant isotope of hydrogen consists of a single proton. In general, an isotope is denoted using the chemical symbol with a left subscript Z and a left superscript A. The chemical symbol and atomic number are redundant, so we’ll drop the subscript. A proton is written as ¹H.

The nucleus to the right of ¹H is ²H, called a deuteron, consisting of one proton and one neutron. Deuterium exists in nature but is rare. The isotope ³H also exists but isn’t stable (its half life is 12 years) so it’s not included in the drawing.

The row above hydrogen is helium, with two stable isotopes: ³He and ⁴He. You probably know the nucleus ⁴He by another name; the alpha particle. As you move up to higher rows you find the elements lithium, beryllium, and  boron (¹⁰B is used for boron neutron capture therapy). Light isotopes tend to cluster around the dashed line Z = N.

Figure 17.2, with part of the drawing magnified.

Moving up and to the right brings us to essential elements for life: carbon, nitrogen, and oxygen. Certain values of Z and N, called magic numbers, lead to particularly stable nuclei: 2, 8, 20, 28, 50, 82, and 126. Oxygen is element eight, and Z = 8 is magic, so it has three stable isotopes. The isotope ¹⁶O is doubly magic (Z = 8 and N = 8) and is therefore the most abundant isotope of oxygen.

Figure 17.2, with part of the drawing magnified.

The next drawing shows the region around ⁴⁰Ca, which is also doubly magic (Z = 20, N = 20). It is the heaviest isotope having Z = N. Heavier isotopes need extra neutrons to overcome the Coulomb repulsion of the protons, so the region of stable isotopes bends down as it moves right. The dashed line indicating Z = N won’t appear in later figures; it’ll be way left of the magnified region. Four stable isotopes (³⁷Cl, ³⁸Ar, ³⁹K, and ⁴⁰Ca) have a magic number of neutrons N = 20. Calcium, with its magic number of protons, has five stable isotopes. No stable isotopes correspond to N = 19 or 21. In general, you’ll find more stable isotopes with even values of Z and N than odd.

Figure 17.2, with part of the drawing magnified.

Next we move up to the region ranging from Z = 42 (molybdenum) to Z = 44 (ruthenium). No stable isotopes exist for Z = 43 (technetium); a blank row stretches across the region of stability. As discussed in Chapter 17 of IPMB, the unstable ⁹⁹Mo decays to the metastable state ^99mTc (half life = 6 hours), which plays a crucial role in medical imaging.

Figure 17.2, with part of the drawing magnified.

Tin has a magic number of protons (Z = 50), resulting in ten stable isotopes, the most of any element.

Figure 17.2, with part of the drawing magnified.

As we move to the right, the region of stability ends. The heaviest stable isotope is lead (Z = 82), which has a magic number of protons and four stable isotopes. Above that, nothing. There are unstable isotopes with long half lives; so long that they are still found on earth. Scientists used to think that an isotope of bismuth, ²⁰⁹Bi (Z = 83, N  = 126), was stable, but now we know its half life is 2×10¹⁹ years. Uranium (Z = 92) has two unstable isotopes with half lives similar to the age of the earth, but that’s another story.

If you want to find information about all the stable isotopes, and other isotopes that are unstable, search the web for “table of the isotopes.” Here’s my favorite: https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html.

Oh Happy Day!

Eric Lander

Russ Hobbie and I hope that Intermediate Physics for Medicine and Biology will inspire young scientists to study at the interface between physics and physiology, and to work at the boundary between mathematics and medicine. But what sort of job can you get with such a multidisciplinary background? How about Presidential Science Advisor and Director of the White House Office of Science and Technology Policy! This week President Biden nominated Eric Lander—mathematician and geneticist—to that important position.

Lander is no amateur in mathematics. He obtained a PhD in the field from Oxford, which he attended as a Rhodes Scholar. Later his attention turned to molecular biology and he reinvented himself as a geneticist. He received a MacArthur “genius” grant in 1987, and co-led the human genome project. Now he’ll be part of the Biden administration; the most prominent scientist to hold a cabinet-level position since biological physicist Steven Chu.

I’m overjoyed that respect for science has returned to national politics. As we face critical issues, such as climate change and the covid-19 pandemic, input from scientists will be crucial. I’m especially excited because not only does our new president respect science, but also—as I wrote in a letter to the editor in the Oakland Press last October—the Congressional representative from my own district, Elissa Slotkin, understands and appreciates science. During her fall campaign, I volunteered to write postcards, one of which you can read below.

The last four years have been grim, but the times they are a-changin’. #Scienceisback.

Oh happy day!

Pioneer in Science: Eric Lander -- The Genesis of Genius

https://www.youtube.com/watch?v=IH4rn50arSY&lc=z13awxxhimyzinzc1231zfxwxpq1cnmqz04

Projections and Filtered Projections of a Square

Chapter 12 of Intermediate Physics for Medicine and Biology describes tomography. Russ Hobbie and I write

The reconstruction problem can be stated as follows. A function f(x,y) exists in two dimensions. Measurements are made that give projections: the integrals of f(x,y) along various lines as a function of displacement perpendicular to each line. For example, integration parallel to the y axis gives a function of x,

as shown in Fig. 12.12. The scan is repeated at many different angles θ with the x axis, giving a set of functions F(θ, x'), where x' is the distance along the axis at angle θ with the x axis.

One example in IPMB is the projection of a simple object: the circular top-hat

The projection can be calculated analytically

It’s independent of θ; it looks the same in every direction.

Let’s consider a slightly more complicated object: the square top-hat

This projection can be found using high school geometry and trigonometry (evaluating it is equivalent to finding the length of lines passing through the square at different angles). I leave the details to you. If you get stuck, email me (roth@oakland.edu) and I’ll send a picture of some squares and triangles that explains it.

The plot below shows the projections at four angles. For θ = 0° the projection is a rectangle; for θ = 45° it’s a triangle, and for intermediate angles (θ = 15° and 30°) it’s a trapezoid. Unlike the circular top-hat, the projection of the square top-hat depends on the direction.

The projections of a square top-hat, at different angles.

What I just described is the forward problem of tomography: calculating the projections from the object. As Russ and I wrote, usually the measuring device records projections, so you don’t have to calculate them. The central goal of tomography is the inverse problem: calculating the object from the projections. One way to perform such a reconstruction is a two-step procedure known as filtered back projection: first high-pass filter the projections and then back project them. In a previous post, I went through this entire procedure analytically for a circular top-hat. Today, I go through the filtering process analytically, obtaining an expression for the filtered projection of a square top-hat. 

Here we go. I warn you, there’s lots of math. To perform the filtering, we first calculate the Fourier transform of the projection, C_F(θ,k). Because the top-hat is even, we can use the cosine transform

where k is the spatial frequency.

Next, place the expression for F(θ,x') into the integral and evaluate it. There’s plenty of book-keeping, but the projection is either constant or linear in x', so the integrals are straightforward. I leave the details to you; if you work it out yourself, you’ll be delighted to find that many terms cancel, leaving the simple result 

To high-pass filter C_F(θ,k), multiply it by |k|/2π to get the Fourier transform of the filtered projection C_G(θ,k)

Finally, take the inverse Fourier transform to obtain the filtered projection G(θ,x') 

Inserting our expression for C_G(θ,k), we find

This integral is not trivial, but with some help from WolframAlpha I found

where Ci is the cosine integral. I admit, this is a complicated expression. The cosine integral goes to zero for large argument, so the upper limit vanishes. It goes to negative infinity logarithmically at zero argument. We’re in luck, however, because the four cosine integrals conspire to cancel all the infinities, allowing us to obtain an analytical expression for the filtered projection

We did it! Below are plots of the filtered projections at four angles. 

The filtered projections of a square top-hat, at different angles.

The last thing to do is back project G(θ,x') to get the object f(x,y). Unfortunately, I see no hope of back-projecting this function analytically; it’s too complicated. If you can do it, let me know.

Why must we analyze all this math? Because solving a simple example analytically provides insight into filtered back projection. You can do tomography using canned computer code, but you won’t experience the process like you will by slogging through each step by hand. If you don’t buy that argument, then another reason for doing the math is: it’s fun!

A Portable Scanner for Magnetic Resonance Imaging of the Brain

A Portable Scanner for Magnetic
Resonance Imaging of the Brain
Cooley et al., Nat. Biomed. Eng., 2020

Chapter 18 of Intermediate Physics for Medicine and Biology describes magnetic resonance imaging. MRI machines are usually heavy, expensive devices installed in hospitals and clinics. A recent article by Clarissa Cooley and her colleagues in Nature Biomedical Engineering, however, describes a portable MRI scanner. The abstract states

Access to scanners for magnetic resonance imaging (MRI) is typically limited by cost and by infrastructure requirements. Here, we report the design and testing of a portable prototype scanner for brain MRI that uses a compact and lightweight permanent rare-earth magnet with a built-in readout field gradient. The 122-kg low-field (80 mT) magnet has a Halbach cylinder design that results in a minimal stray field and requires neither cryogenics nor external power. The built-in magnetic field gradient reduces the reliance on high-power gradient drivers, lowering the overall requirements for power and cooling, and reducing acoustic noise. Imperfections in the encoding fields are mitigated with a generalized iterative image reconstruction technique that leverages previous characterization of the field patterns. In healthy adult volunteers, the scanner can generate T1-weighted, T2-weighted and proton density-weighted brain images with a spatial resolution of 2.2 × 1.3 × 6.8 mm³. Future versions of the scanner could improve the accessibility of brain MRI at the point of care, particularly for critically ill patients.

Cooley et al.’s design has four attributes.

1. It’s designed for imaging the head only. Most critical care MRIs are of the brain, so focusing on imaging the head is not as limiting as you might think. By restricting the device to the head they are able to reduce the weight of their prototype to 230 kg (about 500 pounds); not something you could carry in your pocket, but light enough to be transported in an ambulance or wheeled on a cart. The power required, about 1.7 kW, is far less than for a traditional MRI device, so the portable scanner can be operated from a standard wall outlet.
2. The static magnetic field is produced by permanent magnets. Typical MRI scanners create a static field of a few Tesla using a superconductor, which must be kept cold. Cooley et al.’s device avoids cryogenics completely by using room-temperature, permanent neodymium magnets in a Halbach configuration, producing a static magnetic field of 0.08 T. The lower field strength reduces the signal-to-noise ratio, so advanced MRI techniques such as echo-planar, functional, or diffusion tensor imaging are not feasible. However, many emergency MRIs are used to diagnose traumatic brain injury and don’t rely on these more advanced techniques. The Halbach design results in a small magnetic field outside the scanner, which minimizes safety hazards associated with iron-containing objects being sucked into the scanner.
3. The readout gradient is static. In IPMB, Russ Hobbie and I describe how magnetic field gradients are used to map the Larmor frequency to position. Usually the readout gradient of an MRI pulse sequence is turned on and off as needed. By making this gradient static, Cooley and her collaborators eliminate the need for a power supply to drive it. Most MRI pulse sequences require gradients in three directions, and in Cooley et al.’s device the gradients in the other two directions must still be switched on and off in the traditional way. One side effect of the reduced gradient switching is that this MRI scanner is quieter than a traditional device. This may seem like a minor advantage, but try having your head imaged in a typical MRI scanner with its gradient switching causing a deafening racket.
4. Much of the signal analysis is switched from hardware to software. Because of nonlinearities in the gradient magnetic field, traditional Fourier transform algorithms to convert from spatial frequency to position produce artifacts, and iterative methods that correct for these errors are needed.

Cooley et al.’s article fascinated me because of its educational value; the challenges they face force readers to think carefully about the design parameters and limitations of traditional MRI. If you want to learn more about normal MRI scanners, read this article to see how researchers had to modify the traditional design to overcome its limitations. 

Low-cost MRI systems for brain imaging. by Clarissa Cooley.

https://www.youtube.com/watch?v=bZz3-lmWv4I

An Assessment of Illness in U.S. Government Employees and their Families at Overseas Embassies

“An Assessment of Illness in
U.S. Government Employees and
their Families at Overseas Embassies”
(2020) The National Academies Press.

Recently, a National Academies report examined the illness of staff at overseas embassies.

National Academies of Sciences, Engineering, and Medicine. 2020. “An Assessment of Illness in U.S. Government Employees and their Families at Overseas Embassies.” Washington, DC: The National Academies Press. 

The summary of the report begins

In late 2016, U.S. Embassy personnel in Havana, Cuba, began to report the development of an unusual set of symptoms and clinical signs. For some of these patients, their case began with the sudden onset of a loud noise, perceived to have directional features, and accompanied by pain in one or both ears or across a broad region of the head, and in some cases, a sensation of head pressure or vibration, dizziness, followed in some cases by tinnitus, visual problems, vertigo, and cognitive difficulties. Other personnel attached to the U.S. Consulate in Guangzhou, China, reported similar symptoms and signs to varying degrees, beginning in the following year. As of June 2020, many of these personnel continue to suffer from these and/or other health problems. Multiple hypotheses and mechanisms have been proposed to explain these clinical cases, but evidence has been lacking, no hypothesis has been proven, and the circumstances remain unclear. The Department of State (DOS), as part of its effort to inform government employees more effectively about health risks at posts abroad, ascertain potential causes of the illnesses, and determine best medical practices for screening, prevention, and treatment for both short and long-term health problems, asked the National Academies of Sciences, Engineering, and Medicine (the National Academies) to provide independent, expert guidance.

Then, under the heading of “Plausible Mechanisms,” the summary states

The committee found the unusual presentation of acute, directional or location-specific early phase signs, symptoms and observations reported by DOS employees to be consistent with the effects of directed, pulsed radio frequency (RF) energy.

I’m reluctant to disagree with a report from the National Academies. I wasn’t a member of the committee, I wasn’t consulted by them, and I haven’t analyzed the data myself. Moreover, I’m disturbed by the recent tendency of political leaders to ignore science (for example, on the Covid-19 pandemic and climate change), so I hesitate to reject a review by some of the nation’s top scientists and medical doctors. Nevertheless, I don’t find this report convincing. As I’ve written before in this blog, I’m skeptical that radio-frequency or microwave radiation can explain these effects.

The report states that

Low-level RF exposures typically deposit energy below the threshold for significant heating (often called “nonthermal” effects), while high-level RF exposures can provide enough energy for significant heating (“thermal” effects) or even burns, and for stimulation of nervous and muscle tissues (“shock” effects)... While much of the general public discussion on RF biological effects has focused on cancer, there is a growing amount of data demonstrating a variety of non-cancer effects as well, in addition to those associated with thermal heating.

The absence of certain observed phenomena can also help to constrain potential RF source characteristics. For example, the absence of reporting of a heating sensation or internal thermal damage may exclude certain types of high-level RF energy.

Microwaves affect the body if they’re strong enough to heat tissue (like in a microwave oven). But low-level radio-frequency and microwave effects are less well established (to put it politely). I don’t think there is a growing amount of reliable data demonstrating effects from RF electromagnetic radiation. The same issue arises when discussing the safety of cellular phone radiation (I fear this report will be used by some to support dubious claims of 5G hazards). The report twice cites studies by Martin Pall. As I said previously in this blog “Pall’s central hypothesis is that cell phone radiation affects calcium ion channels, which if true could trigger a cascade of biological effects…. I don’t agree with Pall’s claims.” Neither do Ken Foster and John Moulder (both cited frequently in Intermediate Physics for Medicine and Biology), who wrote that

Despite some level of public controversy and an ongoing stream of reports of highly variable quality of biological effects of RF energy... health agencies consistently conclude that there are no proven hazards from exposure to RF fields within current exposure limits.

The National Academies report emphasizes the Frey effect, in which pulsed electromagnetic radiation causes slight local heating, resulting in tiny transient changes in pressure that can be sensed by the inner ear. In a previous post, I wrote

I am no expert on thermoelastic effects, but it seems plausible that they could be responsible for the perception of sound by embassy workers in Cuba. By modifying the shape and frequency of the microwave pulses, you might even induce sounds more distinct than vague clicks. However, I don’t know how you get from little noises to brain damage and cognitive dysfunction. My brain isn’t damaged by listening to clicky sounds.

The most ridiculous paragraph in the report relates transcranial magnetic stimulation—a technique I worked on while at the National Institutes of Health—to the Frey effect.

If a Frey-like effect can be induced on central nervous system tissue responsible for space and motion information processing, it likely would induce similarly idiosyncratic responses. More general neuropsychiatric effects from electromagnetic stimuli are well-known and are being used increasingly to treat psychiatric and neurologic disorders. In 2008, the Food and DrugAdministration (FDA) approved transcranial magnetic stimulation (TMS) to treat major depression in adults who do not respond to antidepressant medications... Ten years later, the FDA approved office-based TMS as a treatment for obsessive compulsive disorder (OCD)... and portable TMS to treat migraine.

Magnetic stimulation uses large (about 1 Tesla) magnetic fields, and works at low frequencies (1–10 kHz, which are the frequencies that nerves operate at, and are well below the microwave range). Much higher frequencies are unlikely to activate nerves, and such strong magnetic fields will have tell-tale signs. For instance, one way to demonstrate the power of TMS is to place a quarter at the center of a coil and deliver a single pulse. The quarter shoots up into the air, sometimes denting a ceiling tile (I admit, this wasn’t the safest parlor trick I’ve ever witnessed). TMS only activates the brain when the coil is held within a few centimeters of the head. Trying to generate magnetic stimulation remotely would require Herculean magnetic fields that would certainly be noticed (anything metallic would start flying around and heating up). The reason the Frey effect works with weak fields is the extreme sensitivity of the human ear for detecting minuscule pressure oscillations. If the authors of the report think citing transcranial magnetic stimulation is appropriate as evidence to support the plausibility of a Frey-like effect, then they don’t understand the basic physics of how electric and magnetic fields interact with the body.

My view is much closer to that expressed in the article “Havana Syndrome Skepticism” by Robert Bartholomew, which was published in eSkeptic, the email newsletter of the Skeptics Society.

The Frey effect is named after Allan Frey, a pioneer in radiation research. But there are many problems with the explanation. It is highly speculative, and none of the panel members appeared to be experts on the biological impact of microwaves and the Frey effect. Someone who is a specialist on the effect, University of Pennsylvania bioengineer Kenneth Foster, is critical of the report, observing that there is no evidence that the Frey effect can cause injuries. Furthermore, the effect requires a tremendous amount of energy to create a sound that is barely audible... Foster should know, in 1974, he and Edward Finch were the first scientists to describe the mechanism involved in the effect while working at the Naval Medical Research Institute in Maryland...

Foster views any link between his eponymous effect and Havana Syndrome as pure fantasy. “It is just a totally incredible explanation for what happened to these diplomats…. It’s just not possible. The idea that someone could beam huge amounts of microwave energy at people and not have it be obvious defies credibility...” The former head of the Electromagnetics division of the Environmental Protection Agency, Ric Tell, also views the microwave link as science fiction. Tell spent decades working on standards for safe exposure to electromagnetic radiation, including microwaves. “If a guy is standing in front of a high-powered radio antenna — and it’s got to be high, really high — then he could experience his body getting warmer,” Tell said. “But to cause brain-tissue damage, you would have to impart enough energy to heat it up to the point where it’s cooking. I don’t know how you could do that, especially if you were trying to transmit through a wall. It’s just not plausible,” he said.

At the risk of sounding like a grumpy old curmudgeon, I don’t believe the National Academies report. I don’t agree that directed radio-frequency or microwave energy is the most likely explanation for the “Havana Syndrome.” I don’t have a better explanation, but I don’t accept theirs. It’s not consistent with what we know about how electromagnetic fields interact with biological tissue.