Structure of liquids

In one of the twentieth century's greatest didactic works on science, the encyclopedic Course of Theoretical Physics of Landau and Lifshitz, there appears the following statement:

At the time that was written it was indeed true that there was no general theory of liquids – that the only liquid whose properties could be derived from statistical mechanics was liquid helium, and then only in the neighborhood of the absolute zero of temperature. Now, a generation later, the situation has been wholly transformed, and we are able to calculate the properties of ordinary liquids with nearly as much assurance as we do those of dilute gases and harmonic solids.

Nernst heat theorem in thermodynamics

The “third law of thermodynamics,” unlike the first and second laws, does not introduce any new functions of state but rather asserts a universal behavior of the entropy at low temperatures. The most useful form of the third law is the Nernst “heat theorem,” a simplified statement of which is that the entropies of substances approach 0 at the absolute zero of temperature. This statement, which needs much qualification, is explained more fully in this section (a reminder of what one may have learned in thermodynamics), and then its statistical mechanical basis is presented in §5.2.

Suppose a system undergoes a change in its thermodynamic state at a fixed temperature T (isothermal change), and that the accompanying entropy change is ΔST. The subscript reminds us that the entropy change in general depends on the temperature T at which the process occurs. The process could be a chemical reaction, a change of physical state (melting, for example, or transformation between two different crystalline forms), an isothermal expansion or compression, etc. We ask only that the system in both its initial and final states be in a condition of complete internal equilibrium so that it would be possible in principle to carry out the process reversibly (however it may actually be carried out) and so that the system has well defined values of its thermodynamic functions both initially and finally.

This is an introduction to statistical mechanics, intended to be used either in an undergraduate physical chemistry course or by beginning graduate students with little undergraduate background in the subject. It assumes familiarity with thermodynamics, chemical kinetics and the kinetic theory of gases, and quantum mechanics and spectroscopy, at the level at which these subjects are normally treated in undergraduate physical chemistry. Ideas, principles, and formulas from them are appealed to frequently in the present work.

If statistical mechanics constituted about 10% of a physical chemistry course it would be covered in 8 to 12 lectures, depending on whether the course as a whole were taught in two semesters or three. There is enough material in these chapters for 12 lectures (or more). The instructor who has only 8 available will have to be selective. The most technical parts, and so the likeliest candidates for omission or contraction, are the treatment of ortho- and para-hydrogen, which is part of §2.4 of Chapter 2, that of molecular dynamics and Monte Carlo computer simulations in Chapter 7 (§§7.3 and 7.4), and that of the quantum ideal gases in Chapter 8, which includes a discussion of the grand partition function (§8.2).

Because only a relatively short time may be devoted to the subject it is important to arrive quickly at the usable formulas and important applications while still keeping the level consistent with that of an undergraduate physical chemistry course.

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).

Although Brockhouse (1961) originally invented the triple-axis spectrometer for the study of inelastic processes, it has also proven to be an exceptionally useful instrument for measuring elastic scattering in cases where the intensity is inherently low, such as in small single crystals and powders. The three-axis instrument has played an invaluable role in studies of magnetic and structural phase transitions where new superlattice Bragg peaks appear below some critical temperature. By using an analyzer, a better signal-to-noise ratio can be obtained than with the more traditional two-axis instrument with no analyzer. The intrinsic flexibility of a three-axis instrument allows one to vary the incoming and outgoing energy, and this is invaluable in detecting and eliminating the ever-present double scattering occurring within a crystal. The triple-axis instrument has developed into a versatile powder spectrometer because of the ability to significantly change the instrumental resolution by the choice of different analyzer reflections, neutron energies, and collimations.

Three-axis vs. two-axis instruments

Using a triple-axis instrument for measuring elastic scattering can significantly improve the signal-to-noise ratio when compared to the more conventional two-axis instrument used in single-crystal diffractometers or powder spectrometers. The analyzer selects only the scattering that is elastic (within the resolution of the instrument) and the inelastic scattering from the phonons, which occurs at finite energy transfers, is automatically excluded. In a two-axis instrument this phonon scattering always appears as background, and is commonly referred to as thermal diffuse scattering (TDS). Also, since analyzer crystals such as pyrolytic graphite have reflectivities of greater than 50%, there is little intensity lost.