Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

Подождите немного. Документ загружается.

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

Nuclear Magnetic Resonance and the BCS Theory 71

It was important to control the time history of the magnet when we

turned the magnetic ﬁeld oﬀ or on in order to satisfy the adiabatic condition.

Think of the individual nuclear spins as being quantized along the magnetic

ﬁeld each nucleus experiences, either parallel or anti parallel to the eﬀective

magnetic ﬁeld. The crucial time period is when the values of the applied ﬁeld

were comparable to the local ﬁelds, since that was when the nuclear moments

that are quantized along external magnetic ﬁeld change to quantization along

the local ﬁeld. If one turns the ﬁeld to zero too rapidly, the system arrives

at zero ﬁeld with the high ﬁeld magnetization. As Eq. (2) shows, the system

then is not described by a spin temperature since in zero magnetic ﬁeld,

no matter what the spin temperature may be, the magnetization must be

zero. The transition from strong ﬁeld to zero magnetic ﬁeld needs to happen

suﬃciently slowly that the nuclear spin quantization remains at all times

along the vector resultant of the applied and local ﬁelds at each nuclear site.

During the turn-oﬀ of the magnet, adjustment of the time constant of the

turn-oﬀ easily enabled us to satisfy the condition. If we simply reversed the

process when we turned on the magnetic ﬁeld, the fastest rise would occur

at low magnetic ﬁelds, violating the adiabatic conditions. Turning on the

magnet required that we have a process that was slow at low ﬁelds, but

then became fast to avoid relaxation during the turn-on. Figure 9 shows the

scheme we used.

Fig. 9. The actual magnetic ﬁeld cycle. The demagnetization followed an expo-

nential decay. The remagnetization began with a small exponential to carry the H

slowly through values comparable to the local magnetic ﬁeld, from the neighboring

nuclei, then rose rapidly to high values. The resonance was observed on the ﬂy as

it passed through the NMR resonance condition.

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

72 C. P. Slichter

Our sample was “atomized” particles of 99.9% purity obtained from

Alcoa and sieved through a 325-mesh sieve. The mean particle size deter-

mined by microscope was 10 microns. The sample was annealed after many

measurements were made and several of the measurements repeated to look

for eﬀects of strain. DeBye-Scherrer x-ray photographs were taken of the

annealed and unannealed samples to look for the presence of dislocations,

but no diﬀerence in the two photographs could be detected. However, we

did ﬁnd that, after annealing, the sample showed pronounced supercooling

that led to substantial loss in signal because the transition to the supercon-

ducting state was precipitous rather than adiabatic. With this apparatus,

we were able to cool to 0.9 K (a T/T

of 0.8) and we could turn the magnetic

ﬁeld to zero in about 1 msec.

The resonance was observed with a bridgeless system similar to that

used for the thesis of Schumacher. A 400 kHz oscillator of the Pound–

Watkins type

fed the rf through a high impedance to the sample coil,

which was resonated with a parallel capacitance, and fed directly to the

input of an rf ampliﬁer. As used, the apparatus was sensitive to the imagi-

nary part of the nuclear magnetic susceptibily, χ

. As mentioned above, the

rf from the 400 kHz was on continuously, but the NMR resonance condition

was satisﬁed only for the short time that the time-dependent magnetic ﬁeld

passed through the resonance value. Therefore, the NMR signal was a tran-

sient pulse. The signal was then ampliﬁed and displayed on an oscilloscope.

Figure 10 shows an oscilloscope photo of the signal following a single switch-

ing cycle at 1 K. In addition, following the method introduced by Norberg

and Holcomb, we recorded the signal on a gated detector (a boxcar inte-

grator) that enabled us to signal average multiple acquisitions of the signal.

We were able to improve the signal-to-noise ratio by nearly a factor of 3.

Because we had an rf oscillator on during this acquisition, this was phase

sensitive detection, to my knowledge in fact the ﬁrst use of phase sensitive

detection of pulsed NMR signals.

Meanwhile there were important theoretical developments. In 1955,

David Pines and John Bardeen

worked out the theory of the interac-

tion of the conduction electrons and the lattice and how that modiﬁed the

electron–electron coupling. In 1956, Leon Cooper, who succeeded David

as John’s postdoc, worked out the theory of a pair of electrons interacting

above a ﬁlled Fermi sea,

introducing the important ideas of electron pair-

ing and the important role of degeneracy. And Bob Schrieﬀer had started

work as John’s graduate student working on the problem of the origin of

superconductivity.

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

Nuclear Magnetic Resonance and the BCS Theory 73

Fig. 10. Photo of the transient NMR signal observed on the screen of an oscillo-

scope as the magnetic ﬁeld passed through resonance condition. Such signals were

also recorded on a “boxcar integrator” circuit to improve signal to noise ratio.

Fig. 11. David Pines while a postdoc with Bardeen.

We ﬁnished building our apparatus in mid 1956. For our ﬁrst experiments

we studied the relaxation time in zero ﬁeld in the normal state of Al. We

conﬁrmed experimentally the general form of our theoretical prediction of

the magnetic ﬁeld dependence of the nuclear spin-lattice relaxation time in

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

74 C. P. Slichter

Fig. 12. Leon Cooper while a postdoc with Bardeen.

Fig. 13. Bob Schrieﬀer when entering the University of Illinois for graduate study

in Physics.

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

Nuclear Magnetic Resonance and the BCS Theory 75

Fig. 14. Our data: Ratio of the relaxation rate in the superconducting state, R

to its value in the normal state, R

, versus T/T

. The lines are theoretical curves

added later after the creation of the BCS theory.

the normal state and that at all values of applied static magnetic ﬁeld, the

nuclear spin lattice relaxation rate 1/T

, was proportional to T .

We got our ﬁrst experimental results for the superconducting state in the

early fall of 1956. Of course, since there was no broadly accepted theory

of superconductivity, we had no formal prediction of what we would ﬁnd.

But there were approximate ways to think about the problem. A two ﬂuid

model in which there was a normal ﬂuid and a super ﬂuid was popular at

that time. It would appear to imply that the nuclear relaxation would arise

from the normal ﬂuid only and thus the nuclear spin-lattice relaxation rate

would be slower in the superconducting state than in the normal state. We

felt that the same would be true if there were an energy gap since a gap

would inhibit excitations of the electrons need to relax the nuclei.

We were therefore greatly surprised to ﬁnd that the nuclear relaxation

rate (compared with what it would have been for the normal state at that

temperature) increased by a factor of two as we cooled below T

. Our results

are shown in Fig. 14. Several theoretical curves are also shown, but they

came much later.

Chuck and I puzzled about this. We had one possible explanation based

on the Korringa theory and the idea that an energy gap opened up at

the Fermi surface by pushing some states up and others down in energy,

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

76 C. P. Slichter

Fig. 15. Schematic density of states showing a pile-up at the edge of the energy

gap.

increasing the density of states near the edges of the gap. The density of

states then might look like Fig. 15.

As we see from Eqs. (23) and (24), the relaxation rate is determined by an

integral over energy. The energy to ﬂip a nuclear spin is very small compared

with kT . Therefore, we can treat the electron initial and ﬁnal energies as

nearly equal. The factor f(1 −f ) peaks at the Fermi energy, and is zero for

energies more than a few kT away from the Fermi energy. So the relaxation

rate is determined by the integral of the square of the density of states over

this region. If the peaks in Fig. 15 fall within the region, the integral will be

bigger than if the density of states had not been gapped and peaked. The

argument is similar to the inequality 2

+ 0

> 1

+ 1

We went to John to show him our experimental results and asked what he

thought of our explanation. He kindly said that at least it was not foolish!

As we shall see, it turned out later from the BCS theory that this argument is

part of the story, but omits what is the most exciting part of the explanation,

the role of the pairing coherence of the BCS wave function on the matrix

elements!

In October 1956, everyone at Illinois was wildly excited to learn that John,

Walter Brattain, and Shockley would receive the Nobel Prize in Physics for

the invention of the transistor. In December, after his return from Sweden,

John very sweetly told us all about the events surrounding the Prize, even

bringing in the gold medal for us all to see. This was all done without conceit,

but rather the way a dear friend includes one in something meaningful.

Then one morning in early February, I walked out of my oﬃce on the

second ﬂoor of the Physics Building and encountered John. He clearly had

something on his mind and wanted to say something to me. John was not

much of a talker. I realized that if I spoke up, I would preempt what he

wanted to say. So we just stood there while I waited. It seemed like forever,

but was probably just a few seconds. Then he said “Well, I think we have

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

Nuclear Magnetic Resonance and the BCS Theory 77

ﬁgured out superconductivity.” What a message! That was no doubt the

most exciting event in my life as a scientist. Evidently the night before John,

Bob, and Leon had become convinced that they had the solution in hand

and he was coming down the hall to tell me.

In the next few days they sent oﬀ a Letter to Physical Review describing

their concepts.

As yet, there were many predictions to be worked out, but they had

enough results to realize that they had identiﬁed the essence of what caused

superconductivity. In the next few days, they showed Chuck and me their

theory: the Hamiltonian, the wave function, the density of states, how to

write excited states. Chuck and I decided to try to use the theory to calculate

the nuclear spin-lattice relaxation time. The BCS theory was written using

annihilation and creation operators. Neither Chuck nor I had ever used

them. So ﬁrst we set out to use them to solve the relaxation in the normal

state where we knew the answer (the Korringa result). We each did the

calculation. Then we each repeated the calculation using the BCS wave

functions, energy levels, and density of states. To our delight, our answers

agreed with each other! Our result predicted that indeed the relaxation rate

should increase in the temperature region of our data.

We give the details of our calculation in Hebel’s thesis

and in our 1959

paper. I will not repeat it here. The form of the result is an integral very

similar to the conventional result shown in Eq. (23). (Of course, the answer

Hebel and I found was the same as that found by Bardeen, Cooper, and

Schrieﬀer.) However, the integral Σ

is replaced by Σ

(T ) =

∞

)ρ

nmr

, E

, T )f(E

, T )[1−f(E

, T )]dE

, (28)

where the superconducting density of states ρ

obeys

(E) = 0 for |E| < ε

(T )

= ρ

(E)

/[E

− ε

(T )] for |E| > ε

(T ) ,

and C

nmr

, E

, T ) = 1 + [ε

(T )/E

] ,

(29)

with ε

(T ) the energy gap at temperature T , and ρ

(E) the density of states

of the normal metal.

There is an energy gap. The density of states in the superconductor

outside of the gap is seen to be larger than in the normal state, as in Fig. 15.

However, at the edges of the gap, ρ

has an inverse square root inﬁnity. Thus,

the resulting relaxation rate would blow up were we to neglect the nuclear

energy change associated with relaxation and simply set E

= E

. In fact,

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

78 C. P. Slichter

we predicted a tenfold increase in relaxation rate. To bring the theory into

agreement with experiment, we postulated that the electronic energy levels

had a natural width. We postulated that the superconducting states had a

lifetime broadening, and replaced ρ

by ρ

, θ)∆(E

− E)dE

∆(E

− E)dE

= 1 .

(30)

Hebel used a simple rectangular function for ∆. Expressions derived in

this manner are plotted on the data plot. Also included is a point obtained

by Reif

from T

measurements on colloidal mercury using the saturation

method.

In 1957, Morse and Bohm measured sound absorption in superconduct-

ing In and Sn.

Like NMR, absorption of ultrasound involves scattering

of conduction electrons to absorb (emit) the energy when a sound quantum

is emitted (absorbed). In contrast to the NMR result, they found that the

sound absorption rate drops precipitously as one cools below the supercon-

ducting transition. In any one-electron theory, sound absorption and NMR

relaxation should diﬀer primarily in the scale of the interaction strength,

and thus would have the same temperature dependence. Figure 16, based

on a ﬁgure from Cooper’s Nobel Prize lecture,

shows the ultrasound data.

Clearly, the data diﬀer drastically from our NMR result.

It is a major triumph of the Bardeen, Cooper, and Schrieﬀer theory that

it successfully predicts the ultrasound result and thus resolves this paradox.

The essential change is in the integral Σ

. The ultrasound absorption rate

(T ) obeys

(T ) ∝

∞

)ρ

, E

, T )f(E

, T )[1 −f(E

, T )]dE

, (31)

where

, E

, T ) = 1 − [ε

(T )/E

] . (32)

For ultrasound, the factor C

goes to zero as the E

i,f

energies approach

the gap ε

, canceling the inﬁnities from the density of states, whereas for

NMR the factor C

nmr

goes to 2, giving inﬁnities in the density of states

full sway.

If one goes back to the origin of the C factors, one ﬁnds that they are

an expression of the pair nature of the wave function and of an interference

between the state k, spin up and −k, spin down that form the Cooper pairs.

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

Nuclear Magnetic Resonance and the BCS Theory 79

Fig. 16. Data of Morse and Bohm,

for tin and indium, for the temperature

variation of the ratio of the ultrasound absorption in the superconducting state

to its value in the normal state compared. The solid line is the prediction of the

BCS theory.

Thus it is the pair nature of the BCS wave function that contributes two

terms. For NMR the terms are added; for sound absorption one term is

subtracted from the other. Therefore the diﬀerent temperature dependence

of the two experiments is a direct result of the pair nature of the BCS wave

function. The detailed explanation is a feature of Leon Cooper’s Nobel Prize

Lecture.

Although the BCS theory of nuclear relaxation had a peak just below

the superconducting transition, at lower temperatures it predicted that the

relaxation rate would drop exponentially with inverse temperature. We could

not reach such temperatures.

Al Redﬁeld at the IBM Watson Laboratory had become interested

in studying nuclear relaxation in superconductors and had independently

invented the idea of using magnetic ﬁeld cycling for the experiment. He

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

80 C. P. Slichter

Fig. 17. Al Redﬁeld at the controls of his NMR rig while a postdoc with

Bloembergen at Harvard.

learned about our work after Chuck and I were well along on our experi-

ment. Al and I had been good friends for many years. We met during the

war when I worked at the Woods Hole Oceanographic Institution on a war

project. At that time he was in his mid teens. Later we saw each other

at Harvard when I was in graduate school and he an undergraduate. I had

urged him to come to Illinois for graduate study. He did so and in fact did

his PhD thesis using one of the two magnets of my group, measuring the

Hall eﬀect in diamonds. Bob Maurer was his thesis advisor. He then had

gone to postdoc with Bloembergen at Harvard where he worked on NMR,

then to the IBM Watson Lab at Columbia.

He decided to try to extend the NMR data to lower temperatures using

adiabatic demagnetization. He had some results by this method. Then a

major event happened: the price of

He dropped tenfold. Seidel and Keesom

proposed using it as a cryogenic ﬂuid since it did not display superﬂuidity

and thus did not suﬀer from the problems arising from the Rollin ﬁlms.

They demonstrated pumping on

He to cool well below 1 K.

Redﬁeld then

switched to this method and obtained the NMR data well below the tempera-

tures of the peak, verifying the low temperature prediction of our calculation.

His data and ours were displayed in Fig. 18 based on a ﬁgure taken from