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Chapter 7 D. S. Betts, An Introduction to Liquid Helium, 2d ed., Oxford, 1987, chap. 9), and be-
One-Component Phase Equilibrium tween normal conductivity and superconductivity in certain metals. Some metals, for
and Surfaces
example, Hg, Sn, Pb, Al, on being cooled to characteristic temperatures (4.2 K for Hg
at 1 atm) become superconductors with zero electrical resistance.
A lambda transition is one where H T S 0 V at the lambda-point tem-
perature T and C shows one of the following two behaviors: either (a) C goes to
l
P
P
infinity as T is approached from above and from below (Fig. 7.11c), or (b) C goes
l
P
to a very large finite value as T is approached from above and below and the slope
l
C / T is infinite at T (see Prob. 7.44). The shape of the C -versus-T curve resem-
P
l
P
bles the Greek letter l (lambda). Examples of lambda transitions include the transition
4
between liquid helium I and liquid helium II in He; the transition between ferromag-
netism and paramagnetism in metals like Fe or Ni; and order–disorder transitions in
certain alloys, for example, b-brass, and in certain compounds, for example, NH Cl,
4
HF, and CH . (Some people use the term second-order transition as meaning the same
4
thing as higher-order transition.)
4
When liquid He is cooled, as the temperature falls below the lambda temperature
T (whose value depends somewhat on pressure and is about 2 K), a substantial frac-
l
tion of the atoms enter a superfluid state in which they flow without internal friction.
The lower the temperature below T , the greater the fraction of atoms in the superfluid
l
state. (This is a quantum-mechanical effect.) Helium below T is called the helium II
l
4
phase. C of liquid He was measured to within 2 10 9 K of T in a 1992 experi-
l
P
ment on the U.S. space shuttle Columbia, thus eliminating the perturbing effects of
gravity. For the results, see Prob. 7.44.
b-brass is a nearly equimolar mixture of Zn and Cu; for simplicity, let us assume
an exactly equimolar mixture. The crystal structure has each atom surrounded by eight
nearest neighbors that lie at the corners of a cube. Interatomic forces are such that the
lowest-energy arrangement of atoms in the crystal is a completely ordered structure
with each Zn atom surrounded by eight Cu atoms and each Cu atom surrounded by
eight Zn atoms. (Imagine two interpenetrating cubic arrays, one of Cu atoms and one
of Zn atoms.) In the limit of absolute zero, this is the crystal structure. As the alloy is
warmed from T near zero, part of the added energy is used to interchange Cu and Zn
atoms randomly. The degree of disorder increases as T increases. This increase is a
cooperative phenomenon, in that the greater the disorder, the energetically easier it is
to produce further disorder. The rate of change in the degree of disorder with respect
to T increases as the lambda-point temperature T 739 K is approached, and this rate
l
becomes infinite at T , thereby making C infinite at T .
l
P
l
At T , all the long-range order in the solid has disappeared, meaning that an atom
l
located at a site that was originally occupied by a Cu atom at 0 K is now as likely to
be a Zn atom as a Cu atom. However, at T there still remains some short-range order,
l
meaning that it is still somewhat more than 50% probable that a given neighbor of a
Figure 7.12 Cu atom will be a Zn atom (see Fig. 7.12 and Prob. 7.42). The short-range order fi-
Two-dimensional analog of b- nally disappears at a temperature somewhat above T ; when this happens, the eight
l
brass. The upper figure is at atoms that surround a Cu atom will have an average of four Zn atoms and four Cu
absolute zero, where the four atoms. At T , the rate of change of both the short-range order and the long-range order
l
nearest neighbors of each Cu atom with respect to T is infinite.
(shaded) are Zn atoms (unshaded). In solid NH Cl, each NH ion is surrounded by eight Cl ions at the corners of a

The lower figure is at T , where 4 4
l
half the sites that were occupied cube. The four protons of an NH ion lie on lines going from N to four of the eight
4

by Cu atoms at T 0 are now Cl ions. There are two equivalent orientations of an NH ion with respect to the sur-
4

occupied by Zn atoms (and vice rounding Cl ions. At very low T, all the NH ions have the same orientation. As T is
4
versa). However, some short-range increased, the NH orientations become more and more random. This order–disorder

4
order remains at T , in that more transition is a lambda transition. [The situation in NH Cl is complicated by the fact
l
than half the nearest-neighbor 4
pairs contain one Cu atom and one that a very small first-order transition is superimposed on the lambda transition; see
Zn atom. B. W. Weiner and C. W. Garland, J. Chem. Phys., 56, 155 (1972).]