The superconducting state

The discovery, in Section 6.5, of an attractive interaction between electrons in a metal has mentally prepared us for the existence of a phase transition in the electron gas at low temperatures. It would, however, never have prepared us to expect a phenomenon as startling and varied as superconductivity if we were not already familiar with the experimental evidence. The ability to pass an electrical current without any measurable resistance has now been found in a wide range of types of material, including simple elements like mercury, metallic alloys, organic salts containing five- or six-membered rings of carbon and sulfur atoms, and ceramic oxides containing planes of copper and oxygen atoms.

In this chapter we shall concentrate mainly on the simplest type of superconductor, typified by elements such as tin, zinc, or aluminum. The organic superconductors and the ceramic oxides have properties that are so anisotropic that the theories developed to treat elemental materials are not applicable. Accordingly, with the exception of the final Section 7.11, the discussion that follows in this chapter applies only to the classic low-temperature superconductors.

Some of the properties of these materials are shown in Fig. 7.1.1, in which the resistivity, ρ, specific heat, C, and damping coefficient for phonons, α, are plotted as functions of temperature for a typical superconductor. At the transition temperature, Tc, a second-order phase transition occurs, the most spectacular consequence being the apparent total disappearance of resistance to weak steady electric currents.

The Hohenberg–Kohn theorem

In Chapter 2 we explored some of the consequences of electron–electron interactions, albeit in some simple perturbative approaches and within the random phase approximation. There we found that the problem of treating these interactions is exceedingly difficult, even in the case where there is no external one–particle potential applied to the system. We have also explored some of the properties of noninteracting electrons in an external potential, in this case the periodic lattice potential. This led to the concepts of electron bands and band structure, subjects of fundamental importance in understanding the physics of metals, insulators, and semiconductors. Of course, in the real world, electrons in matter are subjected both to electron–electron interactions and to external potentials. How to include systematically and correctly the electron–electron interactions in calculations of real systems is truly a formidable problem.

Why that is so is easily demonstrated. Suppose that we want to solve the problem of N electrons interacting in some external potential. The N-electron wavefunction can be expanded in Slater determinants of some suitable single-particle basis such as plane waves. We can describe the Slater determinants by occupation numbers in our second–quantized notation. Suppose furthermore that we have a basis of a total of Nk plane wave states at our disposal. Here Nk must be large enough that all reasonable “wiggles” of the many-body wavefunction can be included.

The Boltzmann equation

We now have at our command many of the ingredients of the theory of the conduction of heat and electricity. In Section 3.9 we considered the heat current operator for phonons in a lattice, and in Section 4.6 we calculated the velocity of Bloch electrons and their dynamics in applied fields. The missing ingredients of the theory of the transport of heat or electricity, however, are the statistical concepts necessary to understand such irreversible processes. In this chapter we shall adopt the simplest attitude to these statistical problems, and begin with a discussion of the probable occupation number of a given phonon mode or Bloch state.

Let us start by considering a system of independent phonons or electrons. We know that we can define operators nq and nk whose eigenvalues are integers. If, for instance, there are three phonons of wavenumber q′ present then the expectation value 〈nq′〉 of the operator nq′ will be equal to three. If the crystal is not in an eigenstate of the Hamiltonian, however, then 〈nq′〉 may take on some nonintegral value. If we wish to discuss the thermal conductivity of the lattice we should have to interpret the idea of a temperature gradient, and this must certainly involve some departure from the eigenstates of the lattice. We are thus obliged to consider linear combinations of different eigenstates as describing, for example, a lattice with a temperature gradient. Because there will be many different combinations of eigenstates that all give the appearance of a crystal with a temperature gradient we shall only have a very incomplete knowledge of the state of any particular crystal.

The three axes involved in a triple-axis instrument are the monochromator axis, the sample axis and the analyzer axis as shown schematically in Fig. 3.1. However, there are many other elements of a three-axis instrument which are necessary in order to have an efficient spectrometer. These include monochromator and analyzer crystals, energy filters, collimators, and detectors. Since each axis and crystal has to be individually moved for every setting of the instrument, computer control is essential. In addition, extensive shielding is required in order to protect the experimenter and to reduce the overall background or noise in the experiment. In this chapter we shall discuss each of these elements in some detail.

Shielding

Most modern reactors have beam tubes which do not look directly at the core, but instead have their axes tangential to it. This decreases the quantity of unwanted high-energy (≳ 200 meV) neutrons (also called fast neutrons) in the beam tube, since only those neutrons that have a component of velocity nearly parallel to the beam-tube axis can enter the monochromator area. Neutrons which satisfy this condition have necessarily undergone collisions with the moderating material and have lost considerable energy compared to fission energies. The moderation is never perfect, and there are always some fast neutrons that enter the monochromator area, along with γ-rays and other unwanted radiation. It is therefore necessary to shield adequately against this unwanted radiation. The shielding should contain a combination of materials that can scatter the fast neutrons, slow them down, and then absorb them. In addition, a significant amount of lead is needed to absorb the dangerous γ-rays.

The triple-axis spectrometer (TAS) remains arguably the most versatile instrument used for neutron scattering studies, and research using TASs has been of fundamental importance to many areas of condensed-matter science. The power provided by the flexibility of the TAS brings with it some challenges, and determining the optimal conditions for a particular experiment (often referred to as “finding the window”) is a significant one. Various experimental pitfalls have also been identified over the years.

For 32 years (1965–96), triple-axis spectrometers at Brookhaven's High Flux Beam Reactor (HFBR) were the main tools of our Neutron Scattering Group. During that time, many basic techniques for both how and how not to perform experiments were recorded in group research memos. When new post-docs joined the group, they were expected to familiarize themselves with these memos; in addition, they were given a substantial list of references, including numerous research articles, to look up and read. The present book is intended to present the collection of basic techniques in a more-or-less coherent fashion. Besides graduate students and post-docs, we hope that the information presented here will also be useful to experienced researchers who are new to neutron scattering.

Of course, this book would have no purpose if it were not for Bertram Brockhouse and his invention of the TAS. We were extremely pleased when the winners of the Noble Prize in Physics were announced in 1994. The honoring of Brockhouse was long overdue, and it was very appropriate that he shared the prize with Clifford Shull, who did so many of the original experiments demonstrating the power and potential of neutron scattering.

One of the chief strengths of the triple-axis spectrometer is its ability to measure the intensity of scattered neutrons for a particular momentum transfer Q0 and energy transfer ħω0. This capability is especially important for measurements of inelastic scattering from single crystals. With a computer-controlled spectrometer, it is a simple matter to scan the energy transfer while sitting at a specific point in reciprocal space. Conversely, one may choose to scan the spectrometer along a particular direction in reciprocal space while maintaining a constant energy transfer. Such scans are easily interpreted, both qualitatively and quantitatively, as we will discuss.

Of course, because of the small scattering cross section for neutrons and the limited neutron flux generally available, one must typically perform measurements with finite beam divergences and with monochromator and analyzer crystals having significant mosaic widths. As a result, the energy and momentum transfers of the neutrons are distributed within some small region about the average values (ω0, Q0). It is possible to describe the measured signal as a convolution of a spectrometer resolution function R(Q–Q0, ω–ω0) and the scattering function S(Q, ω). The resolution function is peaked at (ω0, Q0) and decreases for deviations (Δω, ΔQ). The constant-amplitude contours for the resolution function form a set of nested ellipsoids in (ω, Q) space and, for a given spectrometer configuration, the volume, shape, and orientation of these ellipsoids depend only on (ω0, Q0). The form of the measured spectra will depend on the way in which the resolution function is scanned through the structures defined by the scattering function. With some basic knowledge about the resolution function one can optimize scans to obtain sharper spectra and a better signal-to-noise ratio.

Among a variety of neutron scattering techniques, those utilizing polarized neutrons have shown rapid development over the last one to two decades. One of the most significant features of using polarized neutrons is the ability to separate magnetic cross sections from nuclear scattering processes. Thus, there exist many important experiments which can be performed only by such techniques. The use of a triple-axis spectrometer with polarization capabilities, which will be our main focus here, makes up a small, but important subset of polarized neutron experiments.

In Chap. 2 we discussed the magnetic scattering cross section for unpolarized neutrons. Here we start with a description of the additional information that can be gained with polarized neutrons. Of course, to take advantage of polarization analysis, one must have techniques for polarizing and manipulating polarized neutron beams, and we give a brief description of these. Turning to applications, we give a few examples of magnetic form-factor measurements in paramagnetic or ferromagnetic systems, which use techniques pioneered by Shull and his collaborators in the 1950s (Shull, Strauser, and Wollan, 1951). Finally, we discuss polarization analysis on a triple-axis spectrometer as developed by Moon, Riste, and Koehler (1969). The latter approach is quite powerful, as it allows one to uniquely isolate both the elastic and inelastic neutron cross sections; however, due to intensity limitations, polarization analysis was not fully utilized until the 1980s. A more extensive review of polarized neutron techniques, including neutron spin-echo spectroscopy, is given in the book Polarized Neutrons by Williams (1988).