Low-Temperature Behavior of Pyroelectricity—A Reply

Abstract

k all three must be proportional to &/ and hence to each other." But unfortunately, since /3 ;J -=f(\kj\) 9 with/(0)=0, satisfies Szigeti's requirements, his argument also implicitly assumes analyticity at wave number k 5 =0. Though the rather numerous assumptions and approximations are otherwise clearly discussed by Szigeti, it is this single, unproven assumption that puts the secondorder dipole terms on the same footing as the anharmonicity, described by b ajj .Although, of course, we have not proved the validity of the molecular-field assumption (1) , 5 we have demonstrated that Szigeti's assumption conflicts with it. On the other hand, if the secondorder dipole terms are set equal to zero in his work from the outset, then his paper gives a correct, microscopic derivation of the rigid-ion result, II x OCT 3^ which was obtained by different arguments in our earlier work. 3 1 B. Szigeti, Phys. Rev. Lett. 35, 1532Lett. 35, (1975). 2 M. Born, Rev. Mod. Phys. JL7, 245 (1945). 3a P. J. Grout and N. H. March, Phys" Lett. 47A, 288 (1974). 3b P. J. Grout, N. H. March, and T. L 0 Thorp, J. Phys. C 8, 2167 (1975). 4 A conceivable way in which this could come about, suggested by Eq. (2.16) of Ref. 3b, is that the coefficient u 2 in Eq.(1) involves the difference ey-e^ between the static and high-frequency dielectric constants in such a manner tha.tu 2^~c o as Co-e^--0, the last result obviously holding in the rigid-ion limit. Such a divergence of u 2 would then herald the presence of a lower-order term in Eq. (1), proportional to \Ap\, in this singular, rigid-ion limit. 5 Indeed, we have shown elsewhere that for a ferromagnet Eq. (1) must be replaced by AU<* (Ap) m where m> 1, Thus, for a ferromagnet we have the rigorous result thatd(AU)/d(Ap) 9 at Ap = 0, is zero. This last result follows also for a pyroelectric if (1) is assumed, whereas if a term ^IA^I is included in (1), as it must be for rigid ions, then this zero derivative condition is violated.The objections by Grout and March against the author's previous paper on pyroelectricity are answered. The earlier statement concerning the behavior of certain expansion coefficients near k = 0 is proved. The conclusions are compared with the results of Born and Huang.In the preceding Comment, Grout and March 1 criticize my recent Letter 2 on pyroelectricity. They say that the molecular-field theory predicts that the pyroelectric coefficient should be proportional to T and not to T 3 at very low temperatures and that my statement concerning the wavenumber dependence of the coefficient fi 5j near k j = 0 is incorrect. It will be shown that both criticisms are erroneous.Since we are discussing the primary pyroelectric coefficient n^ we consider the solid at constant strain. Denote by p(0) the expectation value of the macroscopic dipole moment at T = 0; and by p, the same quantity at temperature T. p(0) contains the contribution of the zero-point vibrations. We may normalize £ and/>(0) to 1 mole of substance. I further define Ap by p =p(0) + Ap. These three quantities have t...

Vorticity dynamics, drift wave turbulence, and zonal flows: a look back and a look ahead

“…When Mima was visiting Hasegawa at Bell Laboratories in 1976, we were introduced to very interesting laser scattering data by Slusher and Surko [1] obtained from density fluctuations of the drift wave frequency range of a Princeton tokamak. The data, in agreement with those obtained by Mazzucato [2] by microwave scattering, showed a very broad frequency spectrum whose width is wider than the expected drift frequency itself. Hasegawa and Mima were intrigued by the data and immediately constructed a theory of drift wave turbulence that may account for this unexpected data.…”

“…At this point the reader should recognize that even if we treat ions in the two-dimensional plane normal to the direction of the magnetic field, we allow electron dynamics in three dimensions here. From equations (2) and 3, we can construct the equation for ion vorticity for its compressible flow,…”

Measurement of the pyroelectric coefficient of poly(vinylidene fluoride) down to 3 K

“…A subsequent version, 10 based on the assumption that the anharmonic term in the potential appears only in p II , predicted a cubic temperature dependence, which is consistent with measurements 25 of pЈ down to 10 K for a crystal of ZnO. A cubic temperature dependence was also predicted when the Born model was subsequently extended to include contributions to p I from the anharmonic potential term as well as the second-order moment, 11,12 arising from deformations of the electron clouds. Neither of these models include possible effects due to the long-range order associated with ferroelectricity.…”

“…Several theoretical models of pyroelectricity have dealt with the low-temperature behavior of p I . [7][8][9][10][11][12][13] One of the early models, developed by Born, used an acoustic-phonon representation 7 to predict a linear temperature dependence for p I in the Debye limit. A subsequent version, 10 based on the assumption that the anharmonic term in the potential appears only in p II , predicted a cubic temperature dependence, which is consistent with measurements 25 of pЈ down to 10 K for a crystal of ZnO.…”

“…There are no previously reported measurements of it below 10 K for PVDF, [1][2][3][4] or for its related copolymer with trifluorethylene, P͑VDF-TrFE͒. 5,6 Predictions by phenomenological models for idealized crystals have differed, historically, between a linear [7][8][9] and a cubic [10][11][12][13] temperature dependence for the primary ͑i.e., strain free͒ component of the pyroelectric coefficient in the low-temperature Debye limit.…”