Lattice defects and specific heats

The subject of this thesis is the measurement and interpretation of specific heats at low temperatures. The measurements fall into two main groups, those on magnetic salts described in Chapters II and III and those on alloys described in Chapters V and VII. The purpose of the former measurements was to obtain information about the energy states of the magnetic ions and the interactions between them and of the latter to investigate the band structure of copper.
A review of the theory of the lattice contribution to the specific heat is given in Chapter I. It is shown that the present state of the theory enables the lattice contribution to be distinguished from any other contribution at the lowest temperatures. However the theoretical calculation of the lattice contribution can still only be carried out to a semiquantitative degree but the general trend of these calculations is towards greater agreement with experimental results.
The apparatus existing when this research was commenced is described in Chapter II. The cryostat consists of an expansion liquefier (Simon, 1932) which carries a vacuum tight radiation shield enclosing the experimental chamber. A calorimeter similar to that described by Wolcott and Smith (1956) is suspended in this experimental chamber.
The only determination of a pure lattice specific heat was carried out on Cerium Magnesium Nitrate. This salt is widely used in obtaining temperatures below 1°K by the method of adiabatic demagnetisation and in measuring those temperatures. Unlike most salts used for demagnetisation experiments, cerium magnesium nitrate has an extremely small magnetic specific heat so that a knowledge of the lattice specific heat is required for a complete understanding of the properties of the salt around 1°K. This was obtained by extrapolation from measurements made in the liquid helium range of temperatures where the magnetic specific heat is negligible compared with the lattice specific heat. It was found that above 3°K the lattice specific heat could not be represented by a single Debye term but below this temperature it is given by CL = 1.13 x 10-3RT-3.
Chapter III is devoted to the study of specific heat anomalies of the Schottky and co-operative types. The results of measurements on a powdered specimen of uranium tetrafluoride which exhibit a Schottky anomaly are given. The lattice specific heat was assumed to be the same as the specific heat of the diamagnetic salt Thorium Tetrafluoride for which specific heat values were available down to 5°K. The maximum of the anomalous specific heat occurs at 6.8°K and its value is 0.245 calories per mole per degree. By a mathematical analysis of the results using a scheme of levels proposed by Bleaney and Wolf (1957) and taking into consideration the investigation of the crystal structure by Burbank (1951) a tentative explanation of the anomalous specific heat is obtained. It is proposed that the observed anomalous specific heat is made up of two simple Schottky anomalies with splittings of the doublet lowest in energy of 14°K and 28.3°K. The present results taken in conjunction with those of Burns, Osborne and Westrum (1958) confirm that there is a difference between the anomalous specific heat of powder and granular specimens of uranium tetrafluoride. It is shown that the anomalous specific heat of the granular specimen can also be accounted for by the same two simple Schottky anomalies the only difference between the granular and powder specimens being the relative number of uranium ions with each energy splitting.
Paramagnetic resonance experiments carried out in this laboratory on dilute and semi-dilute crystals with iridium ions replacing platinum ions in ammonium and potassium chloro-platinates had indicated that there might be antiferromagnetic transitions in the pure chloroiridate salts in the liquid helium range of temperatures. Antiferromagnetic transitions are an example of an order-disorder transition and are therefore accompanied by a λ type anomaly in the specific heat and the presence of such an anomaly provides the most distinct evidence that there is such a transition. The results of measurements of the specific heats of ammonium, potassium and sodium chloroiridates are given which show that an anomaly of the expected λ type occurs at 2.15°K for the ammonium salt, 3.05°K for the potassium salt and 3.95°K for the sodium salt. It was found that these temperatures for the ammonium and potassium chloroiridates correspond to the temperatures at which the susceptibilities show the greatest rate of decrease with temperature. The anomalous entropy of all three salts was found to be of the order of R loge2 as was to be expected for a system with spin 1/2. The shape of the specific heat curve for the sodium salt indicated that the transition for this salt might be more complex than for the ammonium and potassium salts.
The theory of the electronic specific heat of metals is reviewed in Chapter IV and this is followed by a review of both the theory and experimental evidence regarding the electronic band structure of copper Excluding evidence from specific heat measurements it is concluded that the Fermi surface in copper must be assumed to be nearly spherical and contain approximately one free electron per atom. It must also be assumed that the surface is distorted in the (111) directions and almost certainly makes contact with the first Brillouin zone boundaries in those directions.
Methods of investigating the shape of an electronic band are considered especially the method of altering the position of the Fermi level in one element by alloying with another element. The underlying assumption in this method is that the addition of atoms of another metal does not alter the band structure of the solvent metal but only alters the position of its Fermi level.
The results of measurements of the electronic specific heat of copper-zinc alloys which were made to study the shape of the s band in copper are given in Chapter V. The publication by Rayne (1957) of similar measurements on a series of α brasses showed differences between the results of the two researches which lay completely outside the combined limits of accuracy. Repeated measurements of the heat capacity of the same specimen showed that results obtained with the Simon bomb apparatus were not always reproducible and this could be only attributable to helium absorbed by both specimen and calorimeter. It was therefore decided to construct a new apparatus in which the use of helium exchange gas in the calorimeter would be avoided and in which cooling would not be carried out by direct contact with liquid helium.
The new apparatus is described in Chapter VI and may be briefly described as a free helium cryostat containing a vacuum calorimeter which embodies a mechanical heat switch. Its construction, evolution and operation are described in detail. Experimental results obtained with the new apparatus on pure copper and dilute copper-zinc alloys are given in Chapter VII.
The specific heat of pure copper was determined with the dual purpose of testing the apparatus and acting as a reference from which the changes in the electronic and lattice terms could be evaluated The resulting values of γ and θ were:
γ = (0.698 ± 0.002) x 10-3 joules per mole per degree2
θ = 341.3 ± 1.1°K.
These values are compared with other experimental values and values calculated from theory The results are then given for a series of ex brasses none of them containing more than 5% of zinc by weight The initial rise of γ from that for pure copper is shown to be much more steep than indicated by Rayne's results, (1957). There is also evidence of an initial 1% increase in θ on the introduction of zinc atoms into pure copper. The initial rise in γ is inexplicable from the free electron point of view and it must be assumed that the Fermi surface departs appreciably from sphericity. Contrary to the bulk of evidence discussed in Chapter IV it would appear that the Fermi surface makes contact with the zone boundaries at a zinc concentration of about 1%.
The band model proposed by Cohen and Heine (1958) to reconcile this conflicting evidence is then described. These authors propose that the zone boundary effect observed in the specific heat of the ex brasses at low zinc concentrations is not the Fermi surface touching the zone boundaries but pulling away from them. Further evidence for the breakdown of the rigid band model is given and it is concluded that it must be rejected. Whereas a few years ago the abandonment of the rigid band model would have left an uneasy vacuum, the experimental results can now be explained qualitatively with the use of the model of Cohen and Heine until such time as the non-periodic potential problem is finally solved.