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1824
VOPF~- becomes 67.6 f 0.8 which coincides within experimental error with VOPF~-= 67.2 ml mol-l found in KPFs solutions. l9 The Stokes’ radii of AsFB-, PFB-,halide ions, C104-, and Reo4- have been plotted in Figure 1 against their crystallographic radii. All the anions are almost spherically symmetric; however, the points for both fluorinated anions fall off the smooth curve which represents the behavior of all the other anions. This
implies that a different kind of solvent-anion interaction takes place in the fluoro ions, probably due to the presence of fluorine atoms on the surface of these octahedral anions. Acknowledgment. The authors are grateful to the Comisi6n Kacional de Investigaciones Cientificas y Tecnol6gicas (Chile) for financial support. (20) R. H. Stokes and R. A. Robinson, Trans. Faraday Soc., 53, 301 (1957).
C O M M U N I C A T I O N S TO THE E D I T O R Permittivity Measurements in the Time Domain
Sir: Fellner-Feldeggl has recently described a method of measuring the static permittivity KO, the high-frequency permittivity K,, and the relaxation spectrum of a dielectric, by examination of the leading edge of a step function reflected from a sample of the dielectric terminating a coaxial transmission line. The value of K O he presents for the alkyl alcohols are in good agreement with the literature2 but his calculated values of K , and the single relaxation time 7 are not. It is the purpose of this letter to consider why this is so. Eqiiation 3 in the above paper is fundamental to the method of determining the relaxation spectrum. The equation relates the voltage reflection coefficient p from the plane interface of the air and dielectric filled sections of the coaxial line to the permittivity K of the dielectric and is presented as
Whenever dispersion is considered, K should properly be replaced by K * ( u ) = ~ ’ ( u) j t i ” ( ~ )and , p should be represented as a corresponding complex number p*(u). Both K * ( u ) and p * ( w ) can only be defined for a given frequency u, and ey 1 can only be applied to a continuous xave of that frequency. The time variation of the reflected T, oltage resulting from a step function reaching the dielectric interface could be described by a reflection coefficient r ( t ) , and the instantaneous polarization P(t) of the dielectric could be used to define an “instantaneous static permittivity ~ ( t ) ” K(t)
=
4a P(t) __ E(t)
+1
(2)
where E(t) is the applied field. However, since r ( t ) and “ ~ ( t ) ” are defined as transient effects it is quite impermissible to substitute them into eq 1. T h e Journal of Physical Chemistry
The substitution is possible for the times t = 0 and t = m as the response at these times corresponds to the reflection coefficients at infinite and zero frequency (which, being outside the dispersion region, are real numbers). Thus eq 1 is valid for determining K O and K,. As substitution of r ( t ) for p into (1) for any other value of t gives a solution for K which is physically meaningless, it follows that the relaxation times calculated by Fellner-Feldegg on this basis are not the true relaxation times of the dielectrics. Although Fellner-Feldegg’s procedure for determining dispersion data from the reflected step pulse is invalid, the experimental concept presented in the paper must be considered as a very important development in dielectric technique. Point by point permittivity measurements in the frequency domain can be very accurate but invariably demand expensive equipment and are very time consuming. The development of a time domain method would have revolutionary benefits, in terms of speed, measurement ease, and information content, for dielectric work. Comparison of reflected and incident pulse shapes by Fourier transforms would automatically compensate for both irregularly shaped pulses and the frequency response of the detecting system, and would give complete dispersion data at all frequencies up to the limit imposed by signal to noise considerations. The Laplace transform approach advocated by Fellner-Feldegg does not recover the permittivity values of the dielectric inside the dispersion region, and relies rather heavily on the least accurately known parameter of all: p , . Acknowledgment. The author is indebted to the British Empire Cancer Campaign for financial SUPport. (1) H. Fellner-Feldegg, J. Phys. Chem., 73, 3 (1969). (2) E. H. Grant, Proc. Phys. SOC., 70, 937 (1957).
T. A. WHITTINQHAY DEPARTMENT OF RIEDICAL PHYSICS UNIVERSITY OF ABERDEEN FORSTERHILL, ARCRDCEN, SCOTIAND RECEIVED AUGUST15, 1969
