
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.
I. Radio amateur's biography
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. PH
An important
factor was my experience in amateur radio. I spent many hours building my own
transmitters and receivers, learning the code, and getting "on the
air". It helped a lot that my uncle was a radio "ham". In the
process I became fascinated with antennas, transmission lines, standing wave
ratios, and so on. So I grew up as a reader of The Radio Amateur's Handbook and The Radio Amateur's Antenna Book. The great thing was that these
books described the phenomena, giving many important equations with pictures
and discussions, but no derivations. So, one way or another, those had to be
supplied. For example why was the radiation resistance of a halfwave dipole
antenna around 73 ohms? How did the "matching" arrangements (based on
the transmission line equations) work mathematically? To see this operationally
I built two different "directional couplers", and in the end, worked
out the theory of how the reflection coefficient was calculated. All of that
really got me into Electromagnetism. In getting started with the mathematical
theory I luckedout: I had gotten a job in my high school library reshelving
the science books. And I found one book  an English text from around 1890 with
a title that was something like The
Student's Electricity. It began with magnetism, quantifying things as Gauss
had done. And it was simple. For example it derived the BiotSavart Law by
considering the interaction of a current loop and a "monopole". But
it emphasized real experiments and history. I was amazed to learn  as Maxwell
had put it  that the speed of light could be obtained experimentally from the
ratio of the electrostatic measure of a given amount of charge to its
electromagnetic measure. All this proceeded from engineering questions. It was
a very powerful education.
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? PH
In 1947,
graduating in 1951.
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 retaught (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 CauchyRiemann 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
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 deltaprocess 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 BiotSavart 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
wasand still isgreat 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 firstorder 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 semicircle 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
availablebased on just such an insightfor 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, autocorrelation
functions, calculating NMR relaxation rates and linewidths, aspects of the
fluctuationdissipation theorem, and so on. This way the course was really
valuable. It would have a big impact later, on my thesis.
II. The making of a teacher
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
selfdiscovery?
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 handson way,
devising experiments and helping them with basic things.
Then, again
through Professors Holton and Watson, I learned that for the next academic year
(195758), 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 handson 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 homecalibrated double
slit. The semester ended with a sequence on electromagnetic induction. This
included using a homemade transformer handwound 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 "pluggingin" 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 195758 and again for 195859. 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.
III. Nuclear magnetic resonance and critical phenomena
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
"fadedout" 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 meanfield 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 67.3^{o}K) 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 "slowingdown" 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 turnedoff. 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 switchedoff, 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 slowingdown. It was only the rudimentary
"conventional" picturejust a start on learning about critical
dynamics. Still, it was a very helpful start.
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 pumpedout, 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 "fieldmodulation" 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 berylliumcopper 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 MHz110 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 Kushidatype spectrometer was inductively
coupled to the line. This technique could be practiced by looking at the
electron spin resonance absorption at 77^{o}K, 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 lowtemperature 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 nonequivalent 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
closetogether 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 KH^{2} (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 nonlinearly 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 nonlinear 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 65.56^{o}K. 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 KH^{2}. 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 zerofield Neel
temperature. It described the way the sublattice magnetization grew as the
temperature was lowered in the antiferromagnetic state.
I made a preliminary loglog
plot, finding that Nuzero varied essentially as the cube root of TNT. To make
an analysis that was independent of the value taken for the zero field Neel
temperature TN, I then made a plot of Nuzero^{3} 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 0.5^{o}K.
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 2^{o}K 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.


