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Peter Heller Interview

Interview with Peter Heller, Waltham, Massachusetts, 23 January 2002

Interview conducted and recorded by PoS collaborators: Karl Hall and Silvan S. Schweber. Edited by Alberto MartÃ­nez and Silvan S. Schweber, with additions by Peter Heller.

PoS

Maybe we should start with you telling us a little bit about your background, from the time that you remember getting interested in science, your education, elementary school, high school.

PoS

And where did you go to high school?

PH

At the Fieldston School in Riverdale New York. That's one of the Ethical Culture Schools. I had a wonderful teacher by the name of Augustus Klock. It turns out that Robert Oppenheimer was his first good student, and I was one of his last. He taught both Physics and Chemistry. His influence on me was important. This had to do with his approach to teaching.

PoS

And when did you start high school?

PoS

So in 1951, you knew you wanted to become a physicist?

PH

At that point, I wasn't sure. What interested me was a mixture of physics, electrical engineering, and mathematics, all centered on things you could actually do. And the math aspects were very important. Or maybe I should say how they could be rediscovered and taught. For example, my father had given me a book on astronomy - an old one from about 1907. And in that book it said, "If you wish to determine the linear diameter of a heavenly body, like the moon, from its apparent angular diameter and its distance, take the angular diameter in degrees, multiply by the distance in miles, and then divide by 57.29." Well, that's an interesting number! But what is it? At that point I was taking trigonometry in school. In those days we didn't have calculators. So we used "trig tables". For example, the table gave the tangents of the angles between 0 and 45 degrees running down, and then running up from 45 degrees to 89 degrees, the value for 90 degrees being replaced by a straight line as "undefined". Anyway, opposite 89 degrees was the number 57.29. That same number again! That was too good to be a coincidence! There had to be a reason for it. So I figured this out, starting from scratch. It took me a while. But it was a powerful lesson. So if being a physicist means figuring things out from scratch, I guess that's what I wanted.

PoS

So you come to MIT in 1951, you're an undergraduate, and you know you want to be a physics major.

PH

Actually, I began as an electrical engineering major. But the first few years were largely in common. Looking back, the program was very wisely constructed. What do I mean by that? The key concepts were taught from different viewpoints in different departments. For example, the idea of an exact differential was introduced in mathematics and re-taught (in the context of classical thermodynamics) in the second semester of physics, and again later in physical chemistry. So you really got to understand what it meant to say that the entropy was a function of the thermodynamic state. That was a powerful combination. Then, back in mathematics, we learned complex variables and the meaning of "analyticity" leading to the Cauchy-Riemann equations. So in the end you saw that if f(z) was analytic, then the "differential" f(z)dz was "exact" in both its real and imaginary parts. So then you had an insight into Cauchy's theorem. The math course, taught by George Thomas, author of the calculus text, was an absolute gem. All together many connections were made between important things.

PoS

In the first year?

PH

I guess I'm talking about the first two years. And also I took "advanced standing" in mathematics, skipping a year. There's a story about that. One day I was walking in the hall and I ran into Prof. Douglas, my section instructor for the first term of calculus. He called out "Hey, Heller, I've got a problem for you." This was his problem, which I'll never forget: Suppose you have a steel beam one mile long that is lying on the ground. But it is forced to fit between two fixed concrete abutments, one at each end. Now in the heat of the sun it expands in length by one foot. So it has to "arch" a little. How high does it rise in the center? I certainly had no idea. Anyway he told me to go home and work it out. So I went home, and told him the next time, "You know, I worked out your problem. It's amazing! It rises by 45 feet!" I suspect that at the time we were studying the series expansions for the trig functions and I approximated the curve for the "arched beam" by a circle. But later I saw it more fundamentally: When the beam "arches", its length change is second order in its rise. So, taking it the other way, the rise goes as the square root of the length change. That gives you a very large rise for the given length change. Fundamentally it is because a straight line gives the minimum distance between two points. It's a really great problem. I've given it to many students. Many are surprised, and delighted. You can simulate the effect by putting an 18 inch plastic ruler on the table and pushing its ends together by an eighth of an inch, say. It will rise in the center by almost an inch. Anyway, to end the story, Prof. Douglas recommended that I take advanced standing exams for the next two terms of math, which I did.

PoS

When did you switch to physics?

PH

In the second year, though I continued with some EE courses. The circuit theory course was another gem. You learned to think in a powerful new way. I had to work very hard. There were many nights until 2 am, but I really loved it. Those courses were really marvelous for the students who had the conceptual basis. For those that didn't it could be very tough going. A lot of this had to do with the math. Other students used to come to me for help. I enjoyed teaching from the start. I saw that the logic in the lectures often didn't have much of an impact. For example radian measure was introduced and the students were shown that the derivative of the sine was the cosine. Much later I saw that when I asked students to use calculus to estimate the difference between the sines of 30 and 31 degrees, they were off by that factor of 57.29! It helps to get hit over the head with that number! I had been lucky.

PoS

In high school, you had already gotten a good grounding in the calculus?

PH

Well, it wasn't part of the curriculum. But I had a classmate who read about it, and told me about what a derivative was, about the "fundamental theorem", and about Taylor series. I didn't understand all this at first, but I went home and slowly figured things out. I'll never forget how delighted I was when I could see from the delta-process that the derivative of x squared was 2x. Then I went on to tackle other power laws, the trig functions, and more complicated things. And I had very strong motivation: I also heard that there were two "induction laws" in electromagnetic theory. We studied the Faraday Law in high school physics. And from The Student's Electricity's account of the Biot-Savart Law, I figured out what the B field of a moving charge had to look like, and asked myself whether you could get that by differentiating the electric flux through a loop, assuming the charge carried its field lines along with it as it moved. I figured that one out. Later I learned that this picture only worked when v was small compared to c. Still, that problem did a lot for me.
With that background I bought several books as a high school senior. One was The Classical Theory of Electricity and Magnetism by [Max] Abraham and [Richard] Becker. Another was The Fundamentals of Electric Waves by Hugh H. Skilling. These really got me into vector analysis and Maxwell's Equations. Skilling's book was--and still is--great in its engineering approach to teaching E&M. It really answered my antenna questions!

PoS

So you're now a physics major, and you know you want to be an experimentalist.

PH

Well, I wasn't sure. I loved experimentation. It gave me the focus and the motivation. But it wasn't clear to me that I wanted to do an experimental thesis until much later on.

PoS

So you come to Harvard in 1955?

PH

That's right, in the Fall of 1955. I started taking the usual courses. But I can't tell you much about those. What I remember much better are the informal interactions with people like Wendell Furry. He was absolutely wonderful. He would catch me and say "Hey, Peter, I've got a problem for you!" And his problems were terrific. For example: "A farmer has a horse in a very large barn. But he would like to put the horse outside to graze. He has a 100 foot length of fencing which he can attach to two points outside on the longest wall of the barn. How far apart should he place these, and how should he shape the fence to enclose the maximum area?"
So I went back and worked on it, realizing eventually that the placement and shape must be such that if you choose any point on the fence and draw the lines to the attachment points, the triangle you get has to be a right triangle. If it weren't, you could change the attachment distance and the angle at the chosen point, keeping the shapes between the fence and the outer triangle legs fixed. But that could make a first-order increase in the overall area unless the outer angle was 90 degrees. Once you see this, the rest is an exercise in analytic geometry. The answer is a semi-circle with an attachment distance of 100 ft divided by pi over two.
What I loved about this was the fact that you could do it by this insight without the calculus of variations, but that the calculus of variations was available--based on just such an insight--for harder problems.
It was the combination of the pictorial and formal approaches that so interested me.
The graduate course that had the most impact on me was a Special Topics in Experimental Physics course given by Robert V. Pound. This was in the Fall of 1958 and the "Topic" was the fairly new one of Nuclear Magnetic Resonance pioneered by Nicolaas Bloembergen, Robert Pound and Edwin Purcell. I don't remember the body of the course, but I do remember the end, when professor Pound asked us to read up on a specific new NMR technique, and write a report on it. My experience had been that I learned best when I had a chance to work things out for myself in detail, given the right questions. So I went to ask him if he would be willing instead to give us a final assignment consisting of a set of problems - which could be quite difficult - and where we would be challenged to develop the theoretical aspects. He agreed to this, in fact producing a wonderful set of eight problems. I worked very hard on it, and really got to understand in detail about random variables, auto-correlation functions, calculating NMR relaxation rates and linewidths, aspects of the fluctuation-dissipation theorem, and so on. This way the course was really valuable. It would have a big impact later, on my thesis.

PH

So my natural inclination in learning Physics was more like research - slowly working things out for myself - rather than the standard method of going to lectures, reading the textbook, and doing the assigned problems. Coupled with this was a strong interest in teaching, especially at a basic level. How can one help students to "see" these wonderful things and feel a sense of empowerment and self-discovery?
So when an unusual opportunity to teach arose in the fall of 1957, I went after it. It turned out to be very rewarding, and influenced me a lot for later. This started earlier in discussions with Gerald Holton in Physics, and Fletcher Watson of the Graduate School of Education. That led to my taking a summer job teaching a group of high school teachers. I taught them in a very hands-on way, devising experiments and helping them with basic things.
Then, again through Professors Holton and Watson, I learned that for the next academic year (1957-58), a special N.S.F.-funded program would be bringing a group of about 45 high school teachers of science and mathematics to Harvard. The participants were put into some of the regular undergraduate courses, but special conference and lab sections were arranged. The program director was Professor Edwin C. Kemble, who also gave a special set of physics lectures to the group. I was invited to set up the physics lab for them. A teaching fellowship was arranged for me, and I was given a room at Byerly Hall and the help of a machinist.
The teachers were quite enthusiastic about my experiments, and in the director's report at the end of the year, my lab was singled out as the "best part" of the physics program. I'll describe just a few experiments from the spring semester. One goal was to create hands-on labs that could be reproduced at the home schools. For example we measured the wavelength of various colors of visible light by doing Young's experiment with a homemade and home-calibrated double slit. The semester ended with a sequence on electromagnetic induction. This included using a home-made transformer hand-wound on a stack of iron wires for the "core". The "secondary" winding was connected to a flashlight bulb. This bulb would light when the "primary" was excited by a battery via a connection interrupted by contact with a wire scraped along a file. The bulb didn't light when the primary current was steady or when the iron core was removed. This led later to a similar technique to show electrical resonance and produce radio signals: A coil in parallel with a variable capacitor was excited by a battery. Again the trick was to do this through a contact scraped along a file. A similar resonator, with a neon bulb indicator, was seen to be excited when it was brought nearby, and the two capacitor settings were made equal. The transmitter also produced "tuned static" on a portable radio.
There were many other experiments. Some were more like standard introductory labs. Some were "mathlabs". For example, the number e was obtained to a few percent by observing the height decay of a tall vertical column of water that leaked out at the bottom through a fine capillary tube. I also put on "demonstration shows" for the group. The one with the most impact was a way to determine the speed of light (to around 20%) from electric and magnetic measurements, analyzed using the ideas of special relativity. The teachers got the gist of this, and enjoyed it very much. Professors Kemble and Holton really appreciated it.
But mainly I worked very closely in the lab with the teachers in small groups. Theory was discussed in the lab along with the experiments. I put a lot into those sessions, and felt that a lot was accomplished. It certainly was appreciated.
I also had many discussions with the teachers, finding out that the road to a real understanding could be a very rocky one. For example, at the start of the fall term they came to me for help with basic kinematics problems from their theory course. The textbook treated this by deriving the usual general formulae, with particular situations handled algebraically and by "plugging-in" the numbers. But had they understand the derivations? In several cases our discussions showed that they hadn't, and that this was due to major misconceptions. Then I had them go back to the basic principles (what is acceleration?) and work things out numerically at first for simple situations. In other words I had them visualize and document how the object gained or lost its velocity, and where it was at various times. Of course the algebraic formulation had to be there in the end. Many problems couldn't be solved without it. Whenever possible, I got them to supply it, always by asking questions. But that required a lot of hard work.
All this was a great education for me. It told me that the standard instructional method, as important as it was, could fail on its own. You had to supply them with a multitude of questions that were very different from the ones in the book. And they needed a lot of help in sweating through this in detail

PoS

So you spent a couple of years in the NSF program? That was pretty much full time, right?

PH

Well, I did it for the academic year 1957-58 and again for 1958-59. Looking back it seemed like a full time job, because I put so much into it. But, for example, the second year that I did it was the one in which I took Pound's course, which took a lot of time especially at the end. So while the teaching couldn't have been full time, it was a large fraction. In any case it was a labor of love.

PoS

So how did you get to know George Benedek?

PH

The way I remember it happening was that George contacted me. He was very active in looking for students. I suspect that he went around to other faculty, and learned about us. The people who knew me best were Pound and Holton. [In a recent phone call, George has recalled that he learned about me through Jabez Curry Street, who was Chairman of the Physics Department during that period.]

PoS

At Harvard at that stage, did you have to take a lab course? And who taught it?

PH

Yes, I did take a Lab course, taught by Professor Kenneth Bainbridge. While I don't remember the details, I'm sure it included sections on nuclear physics. In any case it wasn't influential in what I did later with George Benedek. That required radio frequency techniques that I had picked up as a radio ham, and dc electrical measurements, for the thermometry, for example, that I was quite comfortable with.

PoS

And what did George tell you about what he wanted you to do?

PH

He outlined what he had done with Dr. [Toshimoto] Kushida on the pressure dependence of the F19 resonance in the antiferromagnet MnF2 at low temperatures. He explained that it gave information on how the pressure affected the hyperfine and superexchange interactions in a combined way. By making measurements at two temperatures, and by making a reasonable assumption in the analysis, the pressure dependence of the superexchange could be found indirectly. But he said that it would be better if this could be done directly by observing how pressure affected the Neel Temperature. Robert G. Shulman and Vincent Jaccarino had already seen the F19 resonance in the paramagnetic state at liquid nitrogen temperature. So that was a known starting point. The question was then whether we could follow the resonance down through the Neel temperature. If this worked, we could learn what we wanted. If, on the other hand, it failed because the lines "faded-out" abruptly as the temperature was lowered, we could use that as an indicator of the Neel point and, by changing the pressure, find out what we wanted.
So George had a very well thought out problem and I was quite impressed. And he knew we were dealing with a critical point where a lot could be learned, depending on how things worked out. For example N.J. Poulis and G.E.G. Hardeman had used the proton NMR in hydrated copper chloride to study antiferromagnetic ordering in the early 1950's. They observed a line splitting proportional to the sublattice magnetization. They even included a plot of the square of the splitting for the region just below the Neel temperature, showing it to have a slope that appeared to diverge as the Neel temperature was reached. This showed dramatically that mean-field theory didn't work. So that was an important experiment historically.
Anyway, going back to MnF2, the pressure dependence aspect seemed like a sure thing. And George mapped out the areas I would be learning about: "You're going to learn a little bit about solid state physics, a little about magnetism. And you're going to learn something about statistical mechanics. You know, this is a phase transition, and not very much is known about phase transitions, so this is an important area."

PoS

Were either of you aware of Michael E. Fisher's calculation in 1959 of the sublattice magnetization?

PH

I myself wasn't. I met Michael Fisher about halfway through my thesis. George had invited Michael to give a seminar. I don't remember the details, but I do remember the excitement of a first exposure to the theory of critical phenomena. Those were the early days, and perhaps it was on that occasion that I first heard about series expansions, Pade approximants, and so on. Those were amazing numerological techniques that gave very impressive working results. I also remember George's prediction to the effect that "some day we'll have a real physical theory".

PoS

What is striking wasn't so much a matter of following the numerology, but that already in 1961 or 1962, you as experimentalists have a sense that maybe getting a critical exponent is something that's worthwhile to do in a laboratory.

PH

We knew how important this was, though again it wasn't the initial goal for my thesis. George had a wonderful sense for what could work, and how to structure the experimental research. Studying the growth of the sublattice magnetization was always a possibility, though how to go about doing that came later. So the first consideration was studying the linewidth in the paramagnetic region. This was found to become very large as the Neel temperature TN (about 67.3oK) was approached from above. That effect was very interesting in itself, and the measurements were excellent preparation for the pressure experiment. So technically my first task was this: Be able to observe the F19 resonance and make linewidth measurements while controlling and measuring the sample temperature to a millidegree.
Later on, there was the problem of getting a physical understanding of this linewidth "anomaly". A theoretical paper on this by Toru Moriya was sent to us from Japan. It seemed like gibberish at the start. But I went to work on it, and in the end, got to make sense out of it. For that, my experience in Pound's course provided background and an important starting point, namely how the NMR linewidth was related to the electron spin correlation functions. But we also needed to understand those quantities. The linewidth anomaly was telling us that the "staggered" spin fluctuations were "slowing-down" as the critical point was reached. I came to a physical understanding of this by imagining that a "staggered field" had been applied and then turned-off. How would the "staggered magnetization" relax? I thought in terms of the "molecular field theory". My essential idea was that the Neel point was the temperature at which a given small staggered magnetization produced just enough "staggered molecular field" to "sustain" that staggered magnetization. So just above the Neel point, the decay should be very slow, since when the applied field was switched-off, a large fraction of the "molecular field" wasn't. This made it possible to relate the decay rate to the staggered susceptibility. There should be an inverse relation. This gave us a physical picture of the critical slowing-down. It was only the rudimentary "conventional" picture--just a start on learning about critical dynamics. Still, it was a very helpful start.
[recording ends]

PoS

Overall, is there something you would like to add? Could you give us an overview of your contribution?

PH

I'll give some details with a timeline indicating the major stages of the work. The first consideration (summer 1959 to late fall 1960) was to develop the temperature control setup with the NMR. In brief, I put the MnF2 sample with its RF coil inside a large thermal mass, in thermal contact with a platinum resistance thermometer and an electrical heater. All of this was inside an "inner dewar" whose jacket could be pumped-out, thermally isolating the sample unit from a pumped liquid (or even solid) nitrogen "bath". I learned the art of controlling the temperature by manually adjusting the heater. The thermal time constants were very long and this worked remarkably well. But the details of this and the NMR needed to be worked out. How could the thermometer's resistance be precisely read using the available Leeds and Northrup K2 potentiometer? What about the effect of the applied field? How could that field be measured? How could the "field-modulation" required by the derivative detection scheme be produced and what were its thermal effects? I studied these problems one by one, and in the end, with many cross checks, found that it was possible to observe the NMR and make all other needed measurements while controlling and reading the sample temperature to better than a millidegree for up to six hours.
In the first set of experiments the large thermal mass was the beryllium-copper high pressure "bomb" designed by George Benedek. Working first at 1 atmosphere, I lowered the temperature toward the Neel point while looking at the F19 NMR with a Pound spectrometer at frequencies below 20 MHz. Under those conditions, the critical linewidth broadening caused the NMR signal to fade into the noise at a temperature that could be pinned down to within 2 millidegrees. This showed that the pressure experiment ought to work. But in advance of that, I made linewidth measurements as a function of temperature, showing an order of magnitude increase in an approximately 3 degree temperature range above the Neel point. That was in January 1961. We presented those linewidth results at the 1961 spring APS Meeting in Washington DC. That stimulated the paper by professor Moriya, and the critical dynamics aspect of the work. Then (mid April to early May 1961), George and I made the measurements to find the pressure dependence of the Neel temperature. The result was more precise than, and in very good agreement with, what Benedek and Kushida had deduced previously by their indirect method.
At this point the initial set of experimental thesis objectives had been met. But the important question remained: Would it be possible to observe the F19 lines just below TN so as to obtain a quantitative account of how the sublattice magnetization increased as the temperature was lowered?
I then reworked the apparatus, replacing the high pressure bomb by a large brass block (slotted to avoid eddy current heating in the NMR modulation field). This block became the large "thermal mass" used to stabilize the temperature. Another technical issue was the need for NMR detection over a wider range of frequencies. The lower end (up to 41 MHz) could be handled with the Pound spectrometer. For the upper end (50 MHz-110 MHz), I found a way to adapt a spectrometer similar to the one used by Kushida. To get this to work with the existing coaxial line down to the sample, I found a trick in which the Kushida-type spectrometer was inductively coupled to the line. This technique could be practiced by looking at the electron spin resonance absorption at 77oK, seen in low applied fields.
The first consideration was simply to locate the F19 resonance in the antiferromagnetic state. Because of the linewidth anomaly at TN, it wasn't possible continuously to follow the resonance down through TN. Also, the cryostat system was such that it wasn't easy to follow the resonance up from its known low-temperature position.
So I decided to try the trick of using a "temperature sweep" over the region just below TN. The point is that in the antiferromagnetic state, changing the temperature was equivalent to varying the "field at the nucleus". I started the first attempt of this kind very late on the evening of June 6, 1961. A weak dc field was applied along the antiferromagnetic axis, and the Pound spectrometer set to about 41 MHz. There were two non-equivalent F19 sites, at which the sublattice contributions to the local field were respectively parallel and antiparallel to the applied dc field. So in the temperature sweep, one should observe two close-together resonances (their separation being determined by the weak dc field). This defined the "signature" of what should be seen using the derivative detection scheme. At somewhat after 1 am on the morning of June 7, I observed this "signature", weekly, but unmistakably, as the temperature was slowly swept through a point approximately 0.65 degrees below the TN. Further runs of this kind confirmed the interpretation. I was so excited that at 5 am I taped a note on the door of George's office. His response the next day was to give the note back to me, saying in effect that I "might want to keep that for the record." I made many more experiments (generally using field sweeps) over the next several months. The results kept improving as the techniques were practiced. By the end of November 1961 there was sufficient data to describe the sublattice magnetization and linewidth behaviors in the critical region below TN.
Fundamentally we were interested in the F19 resonance frequency in zero applied field. (Let's call this Nuzero). But very near TN it was necessary to apply a field to have the resonance appear at frequencies high enough to provide sufficient sensitivity (i.e. above about 25 MHz). So it was important to understand the effects of the applied field on the sublattices.
The applied field H produced a small downward shift of the Neel temperature. This could be found experimentally by a method similar to the one used for the pressure dependence. In brief, I monitored the NMR signal amplitude as the temperature was lowered through the Neel point for two different field values. This was done with the field oriented first along the antiferromagnetic axis (C direction) and then along the A direction. In either case symmetry dictated that the downward shift of TN should go as the square of the field. This gave the coefficient K and the downward shift KH2 (K is positive). This did not exceed 13 millidegrees for any of the measurements made.
To deal with the effect of the field on the sublattices, I made a "model theory" which can be sketched as follows. First, the field should linearly alter the uniform magnetization (the vector sum of the two sublattices) by an amount determined by the known susceptibility. Second, the field should non-linearly alter the "staggered magnetization" (the vector difference of the two sublattices) by an amount that corresponds to the Neel point shift. For example, if this shift were 10 milldegrees, the sample at a given temperature was being brought 10 millidegrees closer to its Neel point. The "model" asserted that, as far as the staggered magnetization was concerned, this was equivalent to a 10 millidegree increase in the temperature.
In particular with the field along the C direction and with T not too close to TN, the non-linear aspect of the "model" predicted that there should exist a downward contribution to the frequency of each of the two F19 sites that varied as the square of the field. I found this to be borne out quantitatively (with the right coefficient) in a detailed set of observations of the NMR lines at 65.56oK. The larger linear term effect could also be determined, and agreed (in addition to the direct effect of the field) with what was calculated from the known susceptibility at several temperatures. I made a corresponding analysis for the case in which the applied field was along the A direction. All this made it possible to deduce accurately what the NMR frequency would have been in zero applied field. In essence, the observed NMR frequencies were used to compute Nuzero from the known linear field effect, and the temperature T was taken as the measured temperature plus the Neel point shift KH2. Additional checks were made on this procedure. The resulting data set (Nuzero vs. T) went from about 5 millidegrees to about 15 degrees below the zero-field Neel temperature. It described the way the sublattice magnetization grew as the temperature was lowered in the antiferromagnetic state.
I made a preliminary log-log plot, finding that Nuzero varied essentially as the cube root of TN-T. To make an analysis that was independent of the value taken for the zero field Neel temperature TN, I then made a plot of Nuzero3 versus T.
This last plot was quite remarkable. Over about the first 2 degrees below TN, the data points scattered almost though not quite evenly (see below) about a drawn straight line. The extension of the line even described the data at 6 degrees below TN to within about 20 millidegrees. However at 14 degrees below TN the discrepancy was 0.5oK. Of course, this had to be the case since the extrapolated critical behavior couldn't possibly work very far below TN.
I then looked more carefully at the data for the first 2 degrees below TN so as to determine what power law worked best. I calculated the effect of plotting Nuzero taken to powers slightly different from 3. I found that the power that gave the straightest plot was 1/0.335. In this case the data scattered evenly and minimally (by a few millidegrees in the temperature) around a drawn straight line. If the power was changed to 1/0.34 or 1/0.33, respectively, the plot showed curvature in one sense or the opposite by several times the minimal scatter. From this I concluded that the critical exponent describing the behavior of Nuzero over this 2oK range had the value beta=0.335 plus or minus 0.005. For our 1962 Phys. Rev. Letter, we doubled this uncertainty, stating the critical exponent as 0.335 plus or minus 0.01.