The Helmholtz Coil and the Maxwell Coil

To do magnetic resonance imaging, you need a static magnetic field that is uniform and a switchable magnetic field that has a uniform gradient. How do you produce such fields? In this post, I explain one of the simplest ways: using a Helmholtz coil and a Maxwell coil.

Both of these are created using circular coils. The magnetic field B_z produced by a circular coil can be calculated using the law of Biot and Savart (see Chapter 8 of Intermediate Physics for Medicine and Biology)

where μ₀ is the permeability of free space (the basic constant of magnetostatics), I is the coil current, N is the number of turns, R is the coil radius, and z is the distance along the axis from the coil center.

The Helmholtz Coil

The Helmholtz coil consists of two circular coils in parallel planes, having the same axis and the same current in the same direction, that are separated by a distance d. Our goal will be to find the value of d that gives the most uniform magnetic field. By superposition, the magnetic field is 

 

To create a uniform magnetic field, we will perform a Taylor expansion of the magnetic field about the origin (z = 0). We will need derivatives of the magnetic field. The first derivative is

(The reader will have to fill in the missing steps when calculating these derivatives.) At z = 0, this derivative will go to zero. In fact, because the magnetic field is even about the z axis, all odd derivatives will be zero, regardless of the value of d.

The second derivative is

At z = 0, the two terms in the brackets are the same. Our goal is to have this term be zero, implying that the second order term in the Taylor series vanishes. This will happen if

or, in other words, d = R. This famous result says that for a Helmholtz pair the coil separation should equal the coil radius.

A Helmholtz coil produces a remarkably uniform field near the origin. However, it is not uniform enough for use in most magnetic resonance imaging machines, which typically have a more complex set of coils to create an even more homogeneous field. If you need a larger region that is homogeneous, you could always just use a larger Helmholtz coil, but then you would need more current to achieve the desired magnetic field at the center. A Helmholtz pair isn’t bad if you want to use only two reasonably sized coils.

The Maxwell Coil

The Helmholtz coil produces a uniform magnetic field, whereas the Maxwell coil produces a uniform magnetic field gradient. It consists of two circular coils, in parallel planes having the same axis, that are separated by a distance d, but which have current in the opposite directions. Again, our goal will be to find the value of d that gives the most uniform magnetic field gradient. The magnetic field is

The only difference between this case and that for the Helmholtz coil is the change in sign of the second term in the bracket. If z = 0, the magnetic field is zero. Moreover, the magnetic field is an odd function of z, so all even derivatives also vanish. The first derivative is

This expression gives us the magnitude of the gradient at the origin, but it doesn’t help us create a more uniform gradient. The second derivative is

This derivative is zero at the origin, regardless of the value of d. So, we have to look at the third derivative.

At z = 0, this will vanish if

or, in other words, d = √3 R = 1.73 R. Thus, the two coils have a greater separation for a Maxwell coil than for a Helmholtz coil. The Maxwell coil would be useful for producing the slice selection gradient during MRI (for more about the need for gradient fields in MRI, see Chapter 18 of IPMB).

Conclusion

Below is a plot of the normalized magnetic field as a function of z for the Helmholtz coil (blue) and the Maxwell coil (yellow). As you can see, the region with a uniform field or gradient is small. It depends on what level of accuracy you need, but if you are more than half a radius from the origin you will see significant deviations from homogeneity.
 

Russ Hobbie and I never discuss the Helmholtz coil in Intermediate Physics for Medicine and Biology. We don’t mention the Maxwell coil by name, but Problem 33 of Chapter 18 analyzed a Maxwell pair even if we don’t call it that.

The Maxwell coil is great for producing the magnetic field gradient dB_z/dz needed for slice selection in MRI, but how do you produce the gradients dB_z/dx and dB_z/dy needed during MRI readout and phase encoding? That, my friends, is a story for another post.

Mr. Clough

A teacher affects eternity; he can never tell where his influence stops. 

Henry Adams

Stephen Clough, from the 1975
Homestead Jr.-Sr. High School Yearbook.

How does someone end up being coauthor on a textbook like Intermediate Physics for Medicine and Biology? It takes a lot of friends, teachers, and role models who help you along the way. I had many excellent teachers when I was young. One of the best was Stephen Clough.

I attended grades 7–10 at Homestead Junior-Senior High School. Usually a junior high and senior high are in separate buildings, but the suburb of Fort Wayne where I lived at the time was new and growing, and had the two combined. For two years (I think grades 9 and 10) I had English with Mr. Clough. He was one of the younger teachers and had longish hair and a mustache, and I thought he was little bit of a hippie. That’s OK, because in the mid 70s hippies were still groovy (although they would go out of fashion soon).

Before I had Mr. Clough, I didn’t read much. I was obsessed with baseball and would read an occasional sports biography, but not much else. I did well in school, but I don’t remember our classes being too challenging or having much homework. Life was about hanging around with friends, playing ping pong, riding bikes, listening to music, and watching television. But Mr. Clough had us reading modern fiction, like Animal Farm and Lord of the Flies. For me, this was an intellectual awakening. Before Mr. Clough I rarely read books; after Mr. Clough I read all the time (and still do).

Me (age 15) from the 1975
Homestead Jr.-Sr. High School Yearbook.

I remember how, on Fridays, Mr. Clough would bring his guitar to school and play for us and sing. I thought this was the coolest thing I’d ever seen. None of my other teachers related to us like that. He played a lot of Dylan. I’ll never forget the day he explained what the words meant in the song American Pie. 

Mr. Clough had a huge influence on my academic development. Reading books led to reading the scientific writing of Isaac Asimov, which led to majoring in physics in college, which led to a PhD, which ultimately led to becoming a coauthor of Intermediate Physics for Medicine and Biology. I owe him much.

As Henry Adams said, a teacher affects eternity. I hope everyone teaching a class using IPMB keeps that in mind. You can never tell where your influence stops. 

I last saw Mr. Clough at my 30^th high school reunion. My friend from high school, Dave Small, became an opera singer, and he sang several songs for us at the gathering. Guess who accompanied him on the guitar? Stephen Clough.

American Pie, by Don McLean.

https://www.youtube.com/watch?v=PRpiBpDy7MQ

J. Robert Oppenheimer, Biological Physicist

J. Robert Oppenheimer.

Did you watch Oppenheimer in the theater this summer? I did. The movie told how J. Robert Oppenheimer led the Manhattan Project that built the first atomic bomb during World War II. But the movie skipped Oppenheimer’s research in biological physics related to photosynthesis.

Russ Hobbie and I only make a passing mention of photosynthesis in Chapter 3 of Intermediate Physics for Medicine and Biology.

The creation of glucose or other sugars is the reverse of the respiration process and is called photosynthesis. The free energy required to run the reaction the other direction is supplied by light energy.

From Photon to Neuron,
by Philip Nelson.

To learn more about Oppie and photosynthesis, I turn to Philip Nelson’s wonderful textbook From Photon to Neuron: Light, Imaging, Vision. His discussion of photosynthesis begins

Photosynthetic organisms convert around 10¹⁴ kg of carbon from carbon dioxide into biomass each year. In addition to generating the food that we enjoy eating, photosynthetic organisms emit a waste product, free oxygen, that we enjoy breathing. They also stabilize Earth’s climate by removing atmospheric CO₂.

Nelson begins the story by introducing William Arnold, Oppenheimer’s future collaborator.

W. Arnold was an undergraduate student interested in a career in astronomy. In 1930, he was finding it difficult to schedule all the required courses he needed for graduation. His advisor proposed that, in place of Elementary Biology, he could substitute a course on Plant Physiology organized by [Robert] Emerson. Arnold enjoyed the class, though he still preferred astronomy. But unable to find a place to continue his studies in that field after graduation, he accepted an offer from Emerson to stay on as his assistant.

Emerson and Arnold went on to perform critical experiments on photosynthesis. Then Emerson performed another experiment with [Charlton] Lewis, in which they found that chlorophyll does not absorb light with a wavelength of 480 nm (blue), but an accessory pigment called phycocyanin does. Emerson and Lewis concluded that “the energy absorbed by phycocyanin must be available for photosynthesis.”

Here is where Oppenheimer comes into the story. I will let Nelson tell it.

Could phycocyanin absorb light energy and somehow transfer it to the chlorophyll system?...

Arnold eventually left Emerson’s lab to study elsewhere, but they stayed in contact. Emerson told him about the results with Lewis, and suggested that he think about the energy-transfer problem. Arnold had once audited a course on quantum physics, so he visited the professor for that course to pose the puzzle. The professor was J. R. Oppenheimer, and he did have an idea. Oppenheimer realized that a similar energy transfer process was known in nuclear physics; from this he created a complete theory of fluorescence resonance energy transfer. Oppenheimer and Arnold also made quantitative estimates indicating that phycocyanin and chlorophyll could play the roles of donor and acceptor, and that this mechanism could give the high transfer efficiency needed to explain the data.

So, what nuclear energy transfer process was Oppenheimer talking about? In Arnold and Oppenheimer’s paper, they wrote

It is the purpose of the present paper to point out a mechanism of energy transfer from phycocyanin to chlorophyll, the efficiency of which seems to be high enough to account for the results of Emerson and Lewis. This new process is, except for the scale, identical with the process of internal conversion that we have in the study of radioactivity.

Internal conversion is a topic Russ and I address in IPMB. We said

Whenever a nucleus loses energy by γ decay, there is a competing process called internal conversion. The energy to be lost in the transition, E_γ, is transferred directly to a bound electron, which is then ejected.

Introductory Nuclear Physics,
by Kenneth Krane.

More detail can be found in Introductory Nuclear Physics by Kenneth Krane.

Internal conversion is an electromagnetic process that competes with γ emission. In this case the electromagnetic multipole fields of the nucleus do not result in the emission of a photon; instead, the fields interact with the atomic electrons and cause one of the electrons to be emitted from the atom. In contrast to β decay, the electron is not created in the decay process but rather is a previously existing electron in an atomic orbit. For this reason internal conversion decay rates can be altered slightly by changing the chemical environment of the atom, thus changing somewhat the atomic orbits. Keep in mind, however, that this is not a two-step process in which a photon is first emitted by the nucleus and then knocks loose an orbiting electron by a process analogous to the photoelectric effect; such a process would have a negligibly small probability to occur.

Nelson compares the photosynthesis process to another process widely used in biological imaging: Fluorescence resonance energy transfer (FRET). He describes FRET this way.

We can find pairs of molecular species, called donor/acceptor pairs, with the property that physical proximity abolishes fluorescence from the donor. When such a pair are close, the acceptor nearly always pulls the excitation energy off the donor, before the donor has a chance to fluoresce. The acceptor may either emit a photon, or lose its excitation without fluorescence (“nonradiative” energy loss).

Let’s put this all together. The donor in FRET is like the phycocyanin molecule in photosynthesis is like the nucleus in internal conversion. The acceptor in FRET is like the chlorophyll molecule in photosynthesis is like the electron cloud in internal conversion. The fluorescence of the donor/phycocyanin/nucleus is suppressed (in the nuclear case, fluorescence would be gamma decay). Instead, the electromagnetic field of the donor/phycocyanin/nucleus interacts with, and transfers energy to, the acceptor/chlorophyll/electron cloud. In the case of FRET, the acceptor then fluoresces (which is what is detected when doing FRET imaging). The chlorophyll/electron cloud does not fluoresce, but instead ejects an electron in the case of internal conversion, or energizes an electron that can ultimately perform chemical reactions in the case of photosynthesis. All three processes are exquisitely sensitive to physical proximity. For FRET imaging, this sensitivity allows one to say if two molecules are close to each other. In photosynthesis, it means the chlorophyll and phycocyanin must be near one another. In internal conversion, it means the electrode cloud must overlap the nucleus, which implies that the process usually results in emission of a K-shell electron since those innermost electrons have the highest probability of being near the nucleus.

There’s lots of interesting stuff here: How working at the border between disciplines can result in breakthroughs; how physics concepts can contribute to biology; how addressing oddball questions arising from data can lead to new breakthroughs; how quantum mechanics can influence biological processes (Newton rules biology, except when he doesn’t); how seemingly different phenomena—such as FRET imaging, photosynthesis, and nuclear internal conversion—can have underlying similarities. I wish my command of quantum mechanics was strong enough that I could explain all these resonance effects to you in more detail, but alas it is not.

Oppenheimer and General Groves
at the Trinity test site. I love
Oppie’s pork pie hat.

If you haven’t seen Oppenheimer yet, I recommend you do. Go see Barbie too. Make it a full Barbenheimer. But if you want to learn about the father of the atomic bomb’s contributions to biology, you’d better stick with From Photon to Neuron or this blog. 

 

 

The official trailer to Oppenheimer.

https://www.youtube.com/watch?v=bK6ldnjE3Y0

 

 

Photosynthesis.

https://www.youtube.com/watch?v=jlO8NiPbgrk&t=14s

The Dobson Unit

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of DNA damage—and therefore cancer—caused by ultraviolet light from the sun. Figure 14.28 in IPMB presents the results of a calculation of UV dose rate, weighted for DNA damage. The caption of the figure states “the calculation assumes clear skies and an ozone layer of 300 Dobson units (1 DU = 2.69 × 10²⁰ molecule m^-2).”

The Dobson Unit, what’s that?

Rather than explaining it myself, let me quote the NASA website about ozone.

What is a Dobson Unit?

The Dobson Unit is the most common unit for measuring ozone concentration. One Dobson Unit is the number of molecules of ozone [O₃] that would be required to create a layer of pure ozone 0.01 millimeters thick at a temperature of 0 degrees Celsius and a pressure of 1 atmosphere (the air pressure at the surface of the Earth). Expressed another way, a column of air with an ozone concentration of 1 Dobson Unit would contain about 2.69 × 10¹⁶ ozone molecules for every square centimeter of area at the base of the column. Over the Earth’s surface, the ozone layer’s average thickness is about 300 Dobson Units or a layer that is 3 millimeters thick.

The Dobson Unit was named after British physicist and meteorologist Gordon Miller Bourne Dobson (1889 –1976) who did early research on ozone in the atmosphere.

Worried about climate change? The ozone story may provide some hope. When man-made chemicals such as chlorofluorocarbons, for example freon, are released into the atmosphere, they damage the ozone layer, allowing larger amounts of ultraviolet radiation to reach the earth’s surface. In the 1970s, an ozone hole developed each year over the south pole. In 1987, countries from all over the world united to pass the Montreal Protocol, which banned many ozone depleting substances. Since that time, the ozone hole has been getting smaller. This represents a success story demonstrating how international cooperation can address critical environmental hazards. Now, we need to do the same thing for greenhouse gases to combat climate change. 

 

How the ozone layer was discovered.

https://www.youtube.com/watch?v=GS0dilngPws

Don't let this happen to your planet!

https://www.youtube.com/watch?v=nCpH71npnvo