References
HebelSlichterRedfield fieldcycling NMR experiment
This experiment was designed [11,12] to measure longitudinal relaxation rate R_{1} (inverse of the relaxation time T_{1 }) of nuclides in both normal and superconducting metals at various applied magnetic fields B (including zero field). Using current terminology, we recognize in it an almost standard [27] Fast FieldCycling NMR Dispersion Relaxometer (FFCNMRD). The original instrument operated in continuous way (CW) as follows (Figure 1):
Figure 1. Timing diagram of the HebelSlichterRedfield experiment
The sample's nuclides are first polarized in a relatively large applied magnetic field (the thin blue line). This is then switched to the desired B_{relaxation} (or just B_{r }) value at which it is left to relax for a variable time τ. The field is then switched to a value which is somewhat higher than the resonance field corresponding to the continuously applied radiofrequency. As the magnetic field flies across the resonance condition one detects a brief, ringing signal S(τ) whose intensity, of course, depends upon τ and also upon the relaxation field value and the absolute sample temperature θ.
Plotting S(τ) as a function of τ, one obtains the relaxation curve for longitudinal magnetization which passes from its equilibrium value in the polarization field to that in the relaxation field. In most metals, this curve is monoexponential and its decay rate coefficient is the measured value of R_{1}(θ,B_{r}).
The modern, pulsed version of the experiment is conceptually similar, but the transmitter and the receiver are both gated, the polarization field can be much higher (there is no interference with nuclear magnetization when the field is switched down) and the signal is acquired at a constant field close to resonance (or even on resonance), giving a wellbehaved free induction decay (FID). The consequent gain in sensitivity and precision is obvious.
While this experimental setup is quite universal, in the case of metals the explanation really applies only to nonsuperconducting samples. Due to the Meissner effect [3,4], the internal magnetic field in a superconductor is rigorously zero even in the presence of an external field (no matter whether static or dynamic). Consequently, whenever the applied field drops below a critical field level, the effective internal field perceived by the sample's nuclides becomes abruptly zero, as indicated by the thick blue line in Figure 1.
Consequently, the only relaxation rate one can measure in a superconductor is that at zero magnetic field.
A few facts about relaxation in normal metals
Figure 2. NMRD profile of a normal metal
Dispersion profiles are best plotted on a loglog scale, as indicated by the labels of the axes. Following usual conventions, however, specific values are indicated without the log in front.
We want to avoid theory here but it is useful to recall at least the phenomenological aspects of NMR relaxation in normal metals which were known [9,10,11] already in 1957, the year when the HebelSlichter effect was discovered.
First of all, the longitudinal relaxation rate R_{1n}(θ,B) in a normal metal is typically proportional to the absolute temperature θ and can be normalized by plotting the ratio r_{1n} = R_{1n}/θ as a function of the applied magnetic field B. The normalized NMR dispersion curve looks typically as shown in Figure 2. There is a plateau at both low and high fields and a dispersion region around some value B_{d} which depends upon the particular metal. According to early theoretical models [10,12], the ratio r_{1n}(0)/r_{1n}(∞) should be equal to 2, but actual experimental values often deviate considerably.
In nonsuperconducting aluminum, for example, the center of the dispersion region is at B_{d} = 1.5 mT, the temperature coefficients are r_{1n}(0) = 2.2 s^{1}K^{1} and r_{1n}(∞) = 0.6 s^{1}K^{1}, their ratio is 3.3, and the maximum loglog slope at the inflex point is 0.6 (the oblique dotted line).
In any case, it is relatively easy to measure NMRD profiles of powdered metal samples, determine their longitudinalrelaxation dispersion parameters, and use the latter to extrapolate the R_{1n}(θ,B) values to any temperature θ and any magnetic field B.
A few facts about superconducting metals
Some metals become superconductors when their absolute temperature θ is lower than a critical value θ_{c} and the external magnetic field B does not exceed some critical value B_{c}(θ) which decreases with increasing temperature and becomes zero at θ_{c}. In a [B,θ] diagram (Figure 3 on the right), the boundary of the region where a metal is in a superconducting state can be approximated quite well by a formula containing just two parameters, the θ_{c} and B_{c} = B_{c}(0).
Figure 3. State diagram of a superconductor
Figure 3 is reminiscent of a phase diagram of a substance, especially when one considers that electric and magnetic field intensities are legitimate thermodynamic state variables. The question is what is the substance, and it is not difficult to guess that it must be the electron gas of the delocalized metal electrons. Superconductivity therefore implies something akin to a phasetransition. That much became clear relatively soon after its discovery by Heike Kamerlingh Onnes [1,2] in 1910, but the nature of the phenomenon remained unclear for nearly half a century. The subsequent history up to  and including  the BCS theory [1621] born in 1957 is well described in Wikipedia, so we can skip it here. Suffice it to say that the BCS theory is based on a microscopic phenomenon  the formation of quasipersistent and quasilocalized electron pairs (Cooper pairs). Since the spins of the two electrons in such a pair are antiparallel, these pseudoparticles have zero total spin and thus behave as doubly charged bosons  something totally different from normal electrons which, being fermions, must obey the rigors of Pauli's exclusion principle. I am of course oversimplifying a complex phenomenon which involves interactions of electrons with phonons and second quantization of the electron field, but the general idea is just that.
Not all metals have a superconducting region and not all superconductors have such a neat diagram. But let us consider a wellbehaved metal like aluminum (B_{c}=9.84 mT, θ_{c}=1.172 °K) and see what its superconductivity diagram implies when it comes to the HebelSlichterRedfield experimental setup of Fig.1. When the field is switched from a high value (well above B_{c }) to near zero, or vice versa, it at some point crosses the critical value B_{c}(θ) indicated in Figure 1 by the dotted blue horizontal line. When changing the sample temperature θ (but staying below θ_{c}), the B_{c}(θ) moves vertically up or down. It is evident from the timing diagram, however, that this does not particularly affect the experiment. So, in principle, R_{1} values can be measured in the superconductive region at any absolute temperature between 0 and θ_{c}.
For completeness, let us say that the dynamics of the normaltosuperconducting transition is very fast and does not interfere with the experiment (at least, nobody has ever complained about it and I don't know whether it has been ever measured).
Figure 4. Experimental HebelSlichter peak
The indices s and n refer to the superconducting and normal state, respectively.
Figure 5. Normalized HebelSlichter effect
HebelSlichter effect
Now we know how to measure the longitudinal relaxation rate R_{1} of a metal's nuclides at any absolute temperature θ and at relaxation field B_{r} = 0 (the latter value is imposed by the fact that in the superconducting state we are not free to set it at will). We even know what to expect of the curve R_{1}(θ) = R_{1}(θ,0) in the normal state (a linear dependence). So what if we now measure R_{1}(θ) at all temperature values, covering also the superconducting region?
The experimental result [11,12,13] is shown schematically in Figure 4. Just below the critical temperature, the relaxation rate abruptly increases above the expected 'normal' value. In the superconducting state, the value of R_{1s}(θ) reaches its maximum at absolute temperature which is about 80% of θ_{c} and then, at temperatures below about 10% of θ_{c}, drops even below the extrapolated 'normal' value R_{1s}(θ) indicated by the dotted line. This R_{1s} temperature profile is known as the HebelSlichter peak.
To account for and, to some extent, eliminate the 'normal' contributions to longitudinal relaxation, it is useful to normalize R_{1}(θ) dividing it by R_{1n}(θ) (either measured or extrapolated). When the absolute temperature is also normalized dividing it by θ_{c}, one obtains a graph like that shown in Figure 5, where the deviation from the normal value of 1 is due entirely to the dynamics of the Cooper pairs. A little bit below θ = θ_{c} the peak reaches a height of almost 3. Notice also that at low values of the ratio θ/θ_{c}, the normalized curve becomes approximately linear.
In practice, the shape of the HebelSlichter peak may deviate considerably from the idealized one shown in Figure 4. In some semiconductors, typically the nonmetallic ones with high critical temperature, it may be even missing altogether. In all cases, however, the shape of the R_{1}(θ) curve, especially of its normalized version, tells us a lot about the formation and dynamics of the Cooper pairs and about the phononic spectrum (lattice vibrations) of the superconducting material.
Historic notes
Longitudinal relaxation times T_{1} and rates R_{1} = 1/T_{1} of nuclides in metals were of interest to physicists since the early days of NMR because they, together with Knight shifts [7,8,9], throw light on the quantum states of electrons with energies lying close to the Fermi level which dominate most of every metal's properties. By late fifties, relaxation phenomena in normal metals were phenomenologically relatively well known and there were reasonably sound theoretical models to describe them.
The HebelSlichter experiment was carried out in 1957 at the University of Illinois by Charles P.Slichter and his pregraduate student L.C.Hebel (sponsored by General Electric). They resorted to the fieldcycling arrangement because that was (and still is) the only way to do NMR in a material which efficiently excludes any magnetic field from its interior. As they amply mention in their two key papers [11,12], they were helped a lot by Alfred Redfield [10,14,15] who was at that time a promising young NMR relaxation expert at the IBM Laboratories, respected both for his theoretical work [10] and for his experiments (he was already oriented towards variablefield NMR relaxometry). This is why I call the experiment HebelSlichterRedfield's, while the effect itself is 'just' HebelSlichter's.
In any case, the actual experimental setup involved a specially build electromagnet with a relatively lowsusceptibility yoke made of thin silicon steel plates, which permitted sufficiently fast, actively driven field switching. They polarized their powder aluminum sample in a field of about 4550 mT and their operating frequency was 400 kHz, corresponding for ^{27}Al to a field of about 36 mT. Aluminum was chosen because both its NMR properties and its superconductivity had been already studied by a number of investigators (including Slichter and Redfield).
The outcome of the experiments was among the key factors in discarding previous theories of superconductivity [5,12] in favor of the new BCS theory which was being concurrently developed at the same University by John Bardeen, Leon Cooper and John Schrieffer. The older theories were simply not able to explain the existence of a local extreme in the R_{1}(θ) curve, while it was possible to slightly adapt the BCS theory to make it match the data (as acknowledged in the HebelSlichter papers [11,12], the two groups were in close contact).
It is interesting that the second set of experiments which cleared the way for the BCS theory regarded ultrasound attenuation [16,20]. Techniques like NMR relaxation, ultrasound absorption and dielectric relaxation are in fact closely correlated since they all reflect the same stochastic processes at atomic (or molecular) level. Combining them should be a rule.
Continuing importance
The historic aspects of the HebelSlicher effect, interesting as they are, should not overshadow the fact that it continues to be an important tool in the study of superconductive materials. This is particularly important for highcriticaltemperature materials were the superconductivity mechanisms are still poorly understood.
It is also proper to mention here that curves remarkably similar to the HebelSlichter peak have been recently measured in lightscattering experiments on superfluids [26]. The parallelism, both theoretical and experimental, is so close that it is not exaggerated to speak about a generalized HebelSlichter effect in many systems whose quantum behavior is due to the formation (condensation) of quasiparticle pairs.
