Ehrenfest Equation Research Articles

Temperature and pressure dependence of thermal expansion coefficient and thermal pressure coefficient for amorphous polymers

The simple P−V relation Vx(P, T) =Vx(P0, T)/[Ax{P + Px(T)}mx] and T−V relation ln T =C[{Vx(P, T)−V(P, 0)}/Vx(P, T)]nx derived in previous works have been used to calculate the compressibility and thermal expansion coefficient by βT = mx/{P + Px(T)} and αp = {Vx(P, T) − V(P, 0)}/{V(P, 0)nxT ln T}. The subscript x means a state of material such as x = g for glass state and x = l for liquid state, mx, Ax and nx are constants and Px(T) is a function of temperature. The Vx(P0, T) is the specific volume at constant pressure P0 and temperature T, and V(P, 0) is the specific volume at pressure P and absolute zero temperature 0 K. The values of mx for 18 amorphous polymers range from 0.0744 to 0.1382, with an average mg = 0.1101 in the glass state and from 0.0709 to 0.1190 with an average mt = 0.0953 in the liquid state. The values of nx for the polymers range from 0.0214 to 0.4526 with an average value ng = 0.1145 in the glass state and from 0.0283 to 0.8700 with an average value nl = 0.1947 in the liquid state. Values of γV have the maximum point against temperature at glass transition temperature (Tg) at which Px(T) decreases rapidly by 1–4 kbar. The Ehrenfest equation at Tg has been examined based on the experimental data of Δβ and Δα at Tg, and it is found that dTg/dP ≠ Δβ/Δα for seven polymers. © 1997 Elsevier Science Ltd.

Wave Packets in a Magnetic Field11The project supported in part by a grant from Chinese Academy of Sciences

A Gaussian wave packet confined to move on a plane perpendicular to a magnetic field remains a Gaussian wave packet in its time evolution. The average position and momentum follow the Ehrenfest equations which are identical to the classical Hamilton equations. A set of nonlinear equations decoupled from the Ehrenfest equation is derived for the parameters describing the time evolution of the density distribution and phases of a wave packet. Explicit solutions are then obtained when the “internal” angular momentum of the wave packet vanishes. In this case it is shown that the motion of the wave packet is a superposition of a translational motion, a rotation and a vibration.

Exact results for a damped quantum-mechanical harmonic oscillator.

Exact calculations for a quantum oscillator linearly coupled to a thermal reservoir are presented. Explicit results (variances and correlations) for the response and the spontaneous fluctuations are evaluated for two different models of the heat bath; a reservoir with a spectrum which is similar to that of acoustic phonons and another which is analogous to optic phonons. The low-temperature behavior is shown to depend sensitively on the spectral density of the thermal reservoir. The mass and the frequency of the oscillator become renormalized through the coupling to the bath. These renormalizations are intrinsically frequency dependent. Moreover, the commonly used form of Ehrenfest's equation for the dynamics of the linearly damped oscillator is shown to only be a reasonable approximation within certain frequency regimes.

Erratum: Validity of the Ehrenfest equation for a system with more than one ordering parameter: Critique of a paper by DiMarzio

Davies and Jones have shown that if the configurational state of a liquid is determined by two or more order parameters that freeze in at the glass transition temperature Tg(P), the Ehrenfest relation, r=ΔβΔCp/TV (Δα)2=1, valid if there is one such parameter, is converted to an inequality r?1. DiMarzio has shown correctly that if the conditions ΔS=0 and ΔV=0 are satisfied at Tg(P), r=1 regardless of the number of order parameters. He has, therefore, concluded that if experimentally r≳1, the order‐parameter concept is inapplicable to the glass transition. It is shown here that r≳1 implies that ΔS and ΔV cannot both be zero at Tg(P), and that there is extensive experimental evidence that the volumes of glasses depend on the pressure under which they were formed, implying ΔV≠0. It is concluded that the experimental observation r≳1 does not render the order‐parameter concept inapplicable.

Validity of the Ehrenfest equation for a system with more than one ordering parameter: Critique of a paper by DiMarzio

Davies and Jones have shown that if the configurational state of a liquid is determined by two or more order parameters that freeze in at the glass transition temperature Tg(P), the Ehrenfest relation, r=ΔβΔCp/TV (Δα)2=1, valid if there is one such parameter, is converted to an inequality r?1. DiMarzio has shown correctly that if the conditions ΔS=0 and ΔV=0 are satisfied at Tg(P), r=1 regardless of the number of order parameters. He has, therefore, concluded that if experimentally r≳1, the order-parameter concept is inapplicable to the glass transition. It is shown here that r≳1 implies that ΔS and ΔV cannot both be zero at Tg(P), and that there is extensive experimental evidence that the volumes of glasses depend on the pressure under which they were formed, implying ΔV≠0. It is concluded that the experimental observation r≳1 does not render the order-parameter concept inapplicable.

High-Magnetic-Field Specific Heat of a Low-Dislocation-Density Alloy Superconductor

The Specific heat $C$ of a well-annealed alloy V-5 at.% Ta, measured at $1.4\ensuremath{\le}T\ensuremath{\le}5$ \ifmmode^\circ\else\textdegree\fi{}K in steady magnetic fields, displays sharp, bulk, superconducting transitions at upper critical fields ${H}_{c2}$ a factor $\ensuremath{\approx}10$ larger than the calorimetrically derived thermodynamic critical fields ${H}_{c}$. The transitions are similar to those observed earlier by Morin ${\mathit{et}\mathit{al}.}^{1}$ in ${\mathrm{V}}_{3}$Ga, but in the present case it is unlikely that the bulk nature of the high-field transitions can be attributed to a nearly complete occupation of the specimen volume by dislocation-centered high-field superconducting filaments of diameter comparable to the penetration depth, since electron transmission microscopy studies on an identically prepared specimen indicate that in at least 95% of the specimen volume the mean separation between dislocations is greater than 1.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ cm. However, the present data are explicable on the basis of the Ginzburg-Landau-Abrikosov-Gor'kov theory with a parameter $\ensuremath{\kappa}\ensuremath{\approx}\frac{{H}_{c2}}{\sqrt{2}{H}_{c}}\ensuremath{\approx}7$. The transition specific heat jumps $\frac{\ensuremath{\Delta}C({T}_{s})}{\ensuremath{\gamma}{T}_{s}}=1.44,1.15,1.10,0.94$ occur at ${T}_{s}=4.30,4.09,3.85,3.37$ \ifmmode^\circ\else\textdegree\fi{}K in fields $H=0,1,2,4$ kG, respectively, where $\ensuremath{\gamma}\ensuremath{\equiv}$ normal state electronic specific heat coefficient = 9.20 mJ/mole ${(\mathrm{K}\mathrm{\ifmmode^\circ\else\textdegree\fi{}})}^{2}$. The $\ensuremath{\Delta}C({T}_{s})$ values are in fair agreement with those calculated via Ehrenfest's equation for second-order phase transitions using Abrikosov's theoretical value of ${(\frac{\ensuremath{\partial}I}{\ensuremath{\partial}H})}_{T}$ at ${T}_{s}$ for $\ensuremath{\kappa}=7$, where $I\ensuremath{\equiv}$ magnetization. For $(\frac{{T}_{s}}{T})\ensuremath{\ge}1.8$, $\frac{{C}_{\mathrm{es}}}{\ensuremath{\gamma}{T}_{s}}=a \mathrm{exp}(\ensuremath{-}\frac{b{T}_{s}}{T})$ with $a=8.95,6.24,5.01,4.7;b=1.48,1.28,1.17,1.1$; for $H=0,1,2,4$ kG, respectively, where ${C}_{\mathrm{es}}$ is the electronic contribution to the specific heat. The exponential temperature dependence of ${C}_{\mathrm{es}}$ down to 1.4\ifmmode^\circ\else\textdegree\fi{}K suggests an essentially everywhere finite, field-dependent, high-field energy gap in accord with Abrikosov's vortex model.