George K. Horton

On the evaluation of equivalent Debye temperatures and related problems

The heat capacity, at constant (0 °K) volume, of face-centred cubic crystals is evaluated by a modified Houston (1948) approximation. There are four principal changes. (a) Instead of three directions in the reciprocal lattice, we use five, namely, the (100), (110), (111), (210) and (211) directions. (b) We replace the first Brillouin zone by a sphere of equal volume. In this way we avoid the difficulty, at low and medium temperatures, of the appearance of the characteristic normalization factor in Houston’s method. At the same time we overcome the difficulty that the latter is only applicable to the integration of functions defined over a sphere. (c) We do not concern ourselves with the frequency spectrum. This is due to a change from integration over frequency to integration over wave-number in the expression for Cv. The reason for the change is that in this way we obtain integrands, in the five directions, that contain no singularities, vary smoothly, and are convenient for numerical evaluation on a digital computer. (d) At higher temperatures we give an expression for Cv in terms of the first three even moments that we prove is in error by less than 0.1 %. We are able to ensure this accuracy by fitting an even polynomial to the Einstein function over the range 0 ≤ hw/kT ≤ 3 to an accuracy better than 0.1 %. Where they overlap the Ɵ(T) curve obtained using the three modifications described earlier lies about 1% below that obtained by the polynomial method. A correction is then applied to eliminate this discrepancy by allowing for the error introduced by replacing the first Brillouin zone by a sphere. The possible error in the final Ɵ(T) curves is shown to be well below 1% for the whole temperature range. We show that the use of more than five directions is not required by comparing the results for five directions with those for six; (221) is the extra direction. The results differ by much less than 1 %. A comparison is also made with earlier work using the conventional three- and five-direction Houston approximations and with the results of Leighton (1948) and Salter (1956). The change to wave-vector integration provides a very convenient technique for passing directly from the dispersion relations, which can be obtained experimentally without reference to a particular model, to various quantities containing averages over the normal mode distribution. The polynomial method differs from that introduced by Thirring (1913) who expanded the Einstein function in a Maclaurin series. We discuss both methods and find that Thirring’s is more suitable for Ɵ, while the polynomial method is more suitable for Cv. Our procedure is applied, by way of illustration, to five representative face-centred cubic metals, Al, Ag, Au, Cu and Pb. We use the nearest-neighbour non-central force theory of Begbie & Born (1947) and the nearest- and next-nearest-neighbour central force theory previously used by Leighton (1948). At each temperature we use the force constants derived from the elastic constants, measured experimentally at that temperature, by the method of long waves. To this extent, at least, we do take account of anharmonic effects. It is found, as was to be expected, that neither theory can quantitatively account for the measured specific heats of the metals studied. Our method may be used in the discussion of related problems such as the thermal expansion of crystals and the thermal conductivity of insulators.

Thermodynamic properties of solid Ar, Kr and Xe based upon a short range central force and the improved self-consistent phonon scheme

Abstract The thermodynamic properties of solid Ar, Kr and Xe are investigated within the framework of the self-consistent phonon method. We have used a central two-body potential of the MieLennard-Jones type. The lowest order self-consistent scheme, the so-called self-consistent phonon harmonic approximation (SC), is shown to be inadequate. The effect of including, in the Helmholtz free energy, the leading terms omitted by the sca is examined in detail and a comparison is also made with our earlier work based on the Born-von Karman approximation. We find substantial differences between the results based on different approximations to the free energy.

Lessons learned: A case study of an integrated way of teaching introductory physics to at-risk students at Rutgers University

In order to provide a physics instructional environment in which at-risk students (particularly women and minorities) can successfully learn and enjoy introductory physics, we have introduced Extended General Physics as an option for science, science teaching, and pre-health professions majors at Rutgers University. We have taught the course for the last five years. In this new course, we have used many elements that have been proven to be successful in physics instruction. We have added a new component, the minilab, stressing qualitative experiments performed by the students. By integrating all the elements, and structuring the time the students invest in the course, we have created a successful program for students-at-risk, indeed for all students. Our aim was not only to foster successful mastery of the traditional physics syllabus by the students, but to create a sense of community through the cooperation of students with each other and their instructors. We present a template for implementation of ou...

A Theoretical Study of the Lattice Dynamics of Neon and Its Isotopes

We present a computationally tractable version of Choquards second-order self-consistent phonon theory and apply it by calculating selected thermodynamic properties of crystalline natural Ne and Ne22 using a Mie-Lennard-Jones nearest-neighbor central force. The model, which is shown to retain the leading correction to the first-order self-consistent theory, gives a quantitative account of the thermodynamics except near the melting point.

Concerning Scientific Discourse about Heat

We aim to examine communication in physics from a linguistic perspective and suggest a theoretical viewpoint that may enable us to explain and understand many physics students’ alternative conceptions. We present evidence, in the context of the concept of heat, that physicists seem to speak and write about physical systems with a set of one or more systematic metaphors that are well understood in their community. We argue that physics students appear to be prone to misinterpreting and overextending the same metaphors and that these misinterpretations exhibit themselves as students’ alternative conceptions. We will analyze physicists’ discourse about heat and present evidence of a connection between students’ alternative conceptions and the possibility that they are misinterpreting the language that they read and hear.

Study of a simple model ferroelectric using the effective potential method

Abstract We have applied the effective potential formalism to a simple one-dimensional model of a ferroelectric crystal. The method incorporates quantum effects into a partition function of classical form, which greatly simplifies the calculation. Our results for the nature of the phase transitions, the transition temperatures, and the temperature dependence of the ordering parameter, are all in very good agreement with a full quantum mechanical calculation. Pacs 63.70, 77.80, 64.60.C

Variational path-integral theory of quantum crystals☆

Abstract We present a theory of the thermodynamics of solids which, for the first time, successfully accounts quantitatively for the entire temperature range without perturbation expansion. Quantum effects are absorbed into an effective potential using a variational path-integral method. This potential is used in a classical Monte Carlo calculation.

Two-phonon Raman scattering of light in rare gas crystals

In this paper we calculate the two-phonon efficiency for light scattering from rare gas crystals at all temperatures. We use a Lennard-Jones potential and the Lorentz-Lorenz model following previous theoretical work and use the first-order self-consistent phonon theory, supplemented by a simple model giving the phonon anharmonic widths and shifts. Our results, though larger than the experimental results, agree well with earlier calculations at 0 K and related molecular dynamics work. The calculated spectra show a rich structure that varies with temperature. We suggest that further experiments could lead to significant new insights into the nature of excitations in rare gas solids as a function of temperature.

Soft mode behavior in a one-dimensional model ferroelectric

Abstract We use a moment-expansion method for the spectral functions to show that a one-dimensional molecular field model of a ferroelectric crystal has a soft zone-center mode of vibration. The frequency of the mode goes towards zero at the transition point but the mode becomes overdamped very close to the transition. Within the limitations of the moment-expansion method, we see no central peak behavior.

An improved self consistent phonon theory for neon

Abstract Thermodynamic properties of solid Ne are evaluated using an improved self-consistent phonon theory that explicitly includes odd derivatives of the potential. The T dependence of thermodynamic properties of Ne are in good agreement with experiment (except for the isothermal compressibility near the melting point — a difficult quantity to measure!) and differ substantially from the predictions of the lowest order theory.