Lambda phase transition

Helium Purification and Liquefaction. HeHum, which is the lowest-boiling gas, has only 1 degree K difference between its normal boiling point (4.2 K) and its critical temperature (5.2 K), and has no classical triple point (26,27). It exhibits a phase transition at its lambda line (miming from 2.18 K at 5.03 kPa (0.73 psia) to 1.76 K at 3.01 MPa (437 psia)) below which it exhibits superfluid properties (27). 

Liquid helium-4 can exist in two different liquid phases liquid helium I, the normal liquid, and liquid helium II, the superfluid, since under certain conditions the latter fluid ac4s as if it had no viscosity. The phase transition between the two hquid phases is identified as the lambda line and where this transition intersects the vapor-pressure curve is designated as the lambda point. Thus, there is no triple point for this fluia as for other fluids. In fact, sohd helium can only exist under a pressure of 2.5 MPa or more. 

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

Another type of phase transition is called a lambda transition, because a graph of heat capacity versus temperature for this type of transition resembles the Greek letter X, as shown in Fig. 4. This type of transition is usually associated with a change from an ordered state to a state with some disorder (order-disorder... 

The Ehrenfest17 classification of phase transitions (first-order, second-order, and lambda point) assumes that at a first-order phase transition temperature there are finite changes AV 0, Aft 0, AS VO, and ACp VO, but hi,lower t = hi,higher t and changes in slope of the chemical potential /i, with respect to temperature (in other words (d ijdT)lowerT V ((9/i,7i9T)higherT). At a second-order phase transition AV = 0, Aft = 0, AS = 0, and ACp = 0, but there are discontinuous slopes in (dV/dT), (dH/<)T), (OS / <)T), a saddle point in and a discontinuity in Cp. A lambda point exhibits a delta-function discontinuity in Cp. 

The existence of a critical point in the pressure-volume-temperature (PVT) diagram (actually, a point in the planar PV projection, but a critical line in a three-dimensional representation), a critical point (Curie temperature) in ferromagnetism, a critical point (Neel point) in antiferromagnetism, a critical temperature in superconductivity, and a critical point (lambda point) in liquid 2He4 are physical descriptions of the onset of a sudden macroscopic collective transition. If one approaches the critical point very closely, dimensionless parameters, defined to describe this approach, are common to all these disparate phenomena the approach to criticality, or to a phase transition, are really the same. 

At even lower temperatures, some unusual properties of matter are displayed. Consequently, new experimental and theoretical methods are being created to explore and describe chemistry in these regimes. In order to account for zero-point energy effects and tunneling in simulations, Voth and coworkers developed a quantum molecular dynamics method that they applied to dynamics in solid hydrogen. In liquid helium, superfluidity is displayed in He below its lambda point phase transition at 2.17 K. In the superfluid state, helium s thermal conductivity dramatically increases to 1000 times that of copper, and its bulk viscosity drops effectively to zero. Apkarian and coworkers have recently demonstrated the disappearance of viscosity in superfluid helium on a molecular scale by monitoring the damped oscillations of a 10 A bubble as a function of temperature. These unique properties make superfluid helium an interesting host for chemical dynamics. 

These data contain a broad lambda type transition with a heat capacity peak at 227.5 K. Powers and Blalock (9) measured high temperature enthalpy data for KOH(cr) in both the a and B phases in a Bunsen ice calorimeter. Their enthalpy data are scattered and not precise enough to accurately define the heat capacities for the a phase. Therefore, the selected heat capacities between 298 and 516 K are estimated by graphical extrapolation of the low temperature heat capacity data. Heat capacities for the B phase are from Powers and Blalock (9). 

A surprising result emerging from the quantum simulations [65, 66] of small ( He)jv clusters and the analyses in Sections II.B and ll.C is the manifestation of a well-characterized, broadened, high-order phase transition for small (" He) y clusters (i.e., N = 8) for the superfluid density [155], and N = 32 for the appearance of the lambda transition [65]. An open question pertains to the threshold size of these equations What is the system s smallest size for the exhibition of superfluidity and what are the corresponding phase transitions ... 

Standard liquid helium cryostats can be used at any temperature above the lambda point of helium, 1.5K, and specialised types of cryostat routinely operate as low as 0.05K. For most samples of chemical interest, there is no practical difference between spectra recorded at 4.2K and at 20K (unless phase transitions occur), thus the additional expense and complexity of very low temperatures cannot be justified. 

Figure 5.13 The Heat Capacity of Helium Near the Lambda Transition. The heat capacity appears to become infinite, as in a first-order phase transition, but it rises smoothiy instead of showing a spike at one point as does a first-order phase transition.

Stance see Fig. 2.2. The most striking properties, however, are those exhibited by liquid helium at temperatures below 2.17 K. As the liquid is cooled below this temperature, instead of solidifying, it changes to a new liquid phase. The phase diagram of helium thus takes on an additional transition line separating the two phases into liquid He I at temperatures above the line and liquid He II at lower temperatures. The low-temperature liquid phase, called liquid helium II, has properties exhibited by no other liquid. Helium II expands on cooling its conductivity for heat is enormous and neither its heat conduction nor viscosity obeys normal rules (see below). The phase transition between the two liquid phases is identified as the lambda line, and the intersection of the latter with the vapor-pressure curve is known as the lambda point. The transition between the two forms of liquid helium, I and II, is called the X... 

The heat capacity of a body is the amount of heat required to produce a temperature rise of 1°C. Specific heat is the heat capacity per unit mass, and generally decreases with lowering temperature, falling to zero at 0°K. Anomalies in the specific heat curve can be related to the molecular origins of phase transitions. Beta brass, for instance, exhibits a lambda point at 470°C due to an order-disorder transition. The specific heat anomaly is referred to as a lambda point because of the resemblance to the Greek letter A. 

Table 3.2 lists the optimal values of the interpolation coefficients estimated by Berman and Brown (1987) for the most common oxide constituents of rock-forming minerals. These coefficients, through equations 3.78.1, 3.78.2, and 3.78.3, allow the formulation of polynomials of the same type as equation 3.54, whose precision is within 2% of experimental Cp values in the T range of applicability. However, the tabulated coefficients cannot be applied to phases with lambda transitions (see section 2.8). 

Other phases are then characterised relative to this ground state, using the best approximation to Eq. (6.1) that is appropriate to the available data. For instance, if die electronic specific heats are reasonably similar, there are no lambda transitions and T 6o, then the entropy difference between two phases can be expressed just as a function of the difference in their Debye temperatures (Domb 1958) ... 

Because these transitions are associated with a mechanism in which one phase gradually evolves into the other, they are also often referred to as continuous or cooperative transitions. The terms second order , lambda , and continuous transitions have often been used interchangeably to refer to the same transition even though a true Ehrenfest second-order heat capacity does not have a lambda shape. We shall use the designation continuous transition (in preference to second order or lambda) for all transitions in which the discontinuity occurs in the second derivative of G. 

Once the alkali alkoxide solutions have been added and the dispersed precipitate has been homogenized, the remaining alkoxides are precipitated by addition of methanol to form insoluble methoxides. Care must be taken to avoid segregation during precipitation due to differing solubilities and reactivities between the various metal alkoxides. When the desired precipitate is heated, the A phase is again the first crystalline phase to form at temperatures below 600°C. The lambda-to-beta transition occurs at 700°C. No p" phase formation occurs below 1200°C. 

---