- Email: [email protected]
Practical aspects of time domain reflectometry
Volume 34, number 3
CHEMICAL PHYSICS LETTERS
1 August 1975
PRACTICAL ASPECTS OF TIME DOhMALNREFLECTOlWETRY T.S. CrLAEKSON and G. WILLIAMS Edward Davies Ckmical Laboratories, Abayshvyth, SY23 IN!?, UK
The Univewity
College
of Wales,
Received 23 April 1975
Dielectric relaxation data For n-propanol ze obtained in the timedomtin and in the frequencydomain (vi;l Fourier transformation) using the method of time-domain refledtometry. The frequencydependent dielectric pennittivity was dedated nrrmerially using an exact solution 2nd various approximate methods. The accuracy OF the avzililable procaiures is assessed in the frequency
and time dam-As.
In
(wr/c)w(W)I
recent years there has been an interest in studyrelaxation behaviour using time-domain spectroscopy [l-9]. Of the various methods available that involve the measurement of the scattering coefficient S, (w) of a thin sample contained in a coaxial line terminated by its characteristic impedance, this is of particular interest. The prefent communication comments on the experimental method and on the
ss(w)‘&!
choice
Cole [9]
;
< 1.
(3)
ing dielectric
of method
for treating
the transient
data.
For the system described above, SQ(w) is given, in the usual notation [I], as
s,(o!=P(w)
1 - exp I-(Zwl/c)[E*(U)l 1 - $(w)exp
C-(2iwllc)
procedure.
There are now sev-
eral approximations of eq. (1) for which e*(w) can be solved analytically. Fellner-Feldeg [2] (thin cell method) S, (w) = (iwl/Zc)[l (2c&/c)l
- c*(w)] ,
[E*(W)]
r/21 4 1 .
van Gemert and de Jeu [6] kd
Sri(U)===
1 -E*(o)
1 +iw[l
+E*(0)]/2c’
&(cd/c)4lef(w)P 4 1. s&4 E3(Ccr)-1=21C wll
x =
[E* (o)ll~z~
(1) or graphical
1 -E”(O) fl 1 +(iwl/2c)[l +c*(w)] +$(iw1/2c)%“(w)’
‘So(w)
(4)
(x cothx f i&/c),
(5)
1121
S,(w) is determined experimentally and the complex permittivity E*(W) may be obtained from eq. (1) by a numerical
Claasen and van Gemert [g]
(2)
iwl[E*(ti)]
l/*/c .
Eq. (5) m.ay be expanded (via 3 series expansion of x cothx) which yields eqs. (3) and (4) as successive appro-ximations. ‘The experimental method we have wed was very similar to that of SugSatt and co-workers [4], and employed a Hewlett-Packard 18 1 TDR system (28 ps tunnel diode) used in conjunction with a PDP-12 computer for signal averaging and data-logging. The sampie length was typicalIy 5 mm contained between PTFE beads (see ref. [2]) which had been specially constructed
with respect
to the inner conductor
so
that they did not give rise to spurious reflections. As in ref. [4] two tiansients were measured, one with and-one without the sample in .tbe Line. The reflected signal from the sample was obtained as the difference .-
46!
Volume 34, number 3
CHEMICAL PHYSICS LETTERS
between the two traces [4]. For each trace 1 OGO digital points were recorded. Dielectric parameters could be determined in the time domain [S] or in the frequency domain by means of Fourier transformation. The transformation was carried out using the Samulon meth.od [lo] on an ICL 4130 computer and using programmes developed by Tuxworth [I I]- One important irregularity arising in the data was associated with spurious reflections at the sampler gate. By choosing an optimum arrangement of delay lines between the tunnel diode and the sampler, and the sampler and the sample, only the reflection samplersample-sampler-sample-sampler was evident and it was usually of srnaJ magnitude. There were also slight mis-matches at the E’TFE beaas. One important quantity is the base-line of the transient, and in the present work this was determin:d as an average of 30 datapoints starting at about 100 data-points before the reference step signnal. It was important that the records of time-domain data were accurately matched before subtraction and that the reflected waveform was accurately referred to the same time-origin as the incident pulse. In the present wclrk no external time-marker was used, so in order to identify a point on a record for matching with a corresponding point on another record two methods were used: (a) the point at which the slope of the transient is greatest is identified as a time-reference point, (b) a Ihear extrapolation from the steepest part of a transient to the voltage b=e-line [IO] defies a time-reference point. Referring to fig. 5b of ref. [4], using either referencing method, record C is matched to record D, then after subtraction but before Fourier transformation record C is matched to record E. For Fourier transformation a zero-time point is defmed on C and on E as corresponding points about 100 data-points bsfore the time-reference point. When analysis is made in the time-domain (vide infra) it becomes necessary to define a zero of time on the response record E. If the incident voltage pulse was a perfect step the zero would be at the first data point of curve E above the baseline. Howeve;, for a non-ideal step this zero may need. to be advanced by as much as the rise-time of the pulse:. The difference in timereferencing using methods (a) and (b) was found typically to be one data-point, i.e., = 5 ps. We have made measurements on n-propanol at 293 K ip a cell‘of liquid depth 5 mm and the following
cogsider the-analysisof the transient data.
paragraphi ,‘.
,-.
.’
197.5
For analysis
in the frequency domain the parameter lo![ea(w)]1/2/cl varied from 0.015 atflo’.5 Hz to 2.0 atf= 1O1o-o Hz. If we require that the criteria for the approximations to hold to be such thatx -=@1 should be taken to be x < 0.01 then in eqs. (2), (3) and (4) above we would require Iwf [E*(W)] l/2/ c I to be < 0.005, 0.17 and 0.82 respectively. This means that the latter two approximations would be useful for this system, up to 0.5 GFlz and 3 GHz. respectively. In practice the best procedure is to solve eq. (1) exactly, and the NewtonRaphson iteration method was used in the present work, and was found to be accurate and efficient under conditions such that Iw![E*(w)] lj2/,l < 1.5. However, at higher frequencies or with longer lengths I of dielectric two undesirable effects become apparent. First the fact that eq. (I) may lead to multiple solutions for E* for a given Sl, (w) may mean that the iteration procedure evaluates the incorrect solution to eq. (1). For any dielectric when IcJ[E*(w)] li2/cl e 2.2 there exists a pair of solutions of similar magnitude for E’ for a given S, (w). Second, we have observed that although the experimental data for the range 1.5 < Iwl[e*(w)] I/*/,! < 2.2 appeared quite accurate, reliable values for E*(W) could not be ob-
tained because of an ill-favoured region in the solution of eq. (1) - small changes in S, (0) lead to large changes in E”(O). Figs. la and lb illustrate this particular point. In fig. la reasonably accurate experimental values of the components of 27,(0) give correspondingly acceptable values for the components of P(w). In fig. lb, one may be in an ill-favoured region, and the values oft*, thus determined, could be subject to large errors. We conclude that the solution of the full equation, eq. (I), has the limitation although in theoIwl[e*(w)]l/2/cl < 1.5, in practice, ry there is no such limitation. Thus the extra frequency range or cell length that can be handied by the solution of eq. (l), via the Newton-Raphson or some similar iteration method, is only better than the best approximation [eq. (4)J by about a factor of 2. The
experimenter should plan applications of the present method to I~J[e*(w)]~‘~/cl < 1 S, using eq. (l), or if the criteria for the approximations [eqs. (2)-(4)] are met and computing facilities are more limited, the approximations can be used up to the values of IwI[e*(w)] ‘/!/cl indicated. The major part of t!e
computation will be the numerical Fourier Transformation.
452.’ :
1 Ausst
Volume
34, number
CHEMICAL
3
PHYSICS
1 August
LETTERS
1975
Fk,. 2. 5 ns transient
records for the line terminated by its impedance and containing a 5 mm cell. Tran-
chaacteristic sient
(a) empty
cd,
(b) cdl filled
with n-propanol
at 293 K.
Fellner-Feldegg [2,3] and Cole [S]. In the usual notation [2,3,5] : Fellner-Fe!degg thin.cell method [2,3] r(& r) = -(U2cro)(~o-~,) for
14
Cole
exp(-I/To), t/To
and
2CT&,-3Eml
4
(6)
~CT&O--Em).
method [5]
$(t) f (e--l)
= -(2c/l)
1 {r(t’)/Vo)dr’ 0
6) Fig. 1. Plots ofSU(w) on the complex plane E*(W). Continuous curves repreent Re S2(w) and the orthogonal set of broken curves reprexnts Im 31 (w). (nj w i/c = 0.1, (b) wl/c = 0.8, where there is a possibility of two E* vtilues for a particular S, value. Graphs such as these can be used to solve eq. (1).
A low frequency limit for the Fourier transform aisses from uncertainties in methods [eqs. (l)-(4)] the base-line determination, owing, in part, to droop on the base-line. In our experiments errors were noticeable at frequencies below 107e5 Hz, and positive values of ReS,, (0) tid hence negative values of E” may be obtained. The measurements using a 5 mm cell for n-propanol
together
with eqs. (l)-(4)
may
therefore give acceptable results in the frequency range 107m5to 10gm5Hz. Two approximations are available for the analysis of transient data in the time domain and are due to
i (2c/l) j
(r(t') r(r'-f)/Vg)
dt’-ti(t)jVo
.
(7)
0
We have used eqs. (l)-(4) and eqs. (6) and (7) for an analysis of transient data for n-propznol in a 5 mm cell at 293 K. Fig. 2 shows a 5 ns scan for the transients for the empty and k-tied cell. Table 1 lists E’and E” obtained over the frequency range using the different
methods
of analysis
in the frequency
don&u
the steepest-slope time referencing method. The results from eqs. (2). (3) and (4) all agree with those obtained by solution of eq. (!) at low frequencies, and we see that the results using eq. (4) compare very satisfactorily with those from eq. (1) at all frequencies, in confirmation of Claascn uld van Gemert [8]. Changing the time-reference by one datz-point (= 5 ps) has a large effect on the values for E’and E”
given
c&dated
from the exact
relation,
eq. (1). This is
shown in table I and in fig. 3 - the Argand
dizg-ram.
463
Volti~&
W, number
3
CHEMICAL
Table 1 Values of (E’, l n) at different
frequencies
1 August 1975
by eqs. (l)-(4) Eq. (4)
TRL a)
TR2 b)
5.34 6.35 7.30 8.07 8.60 8.89 8.81 8.21 7.53
19.60, 3.53 19.06,4.32 18.31,5.13 17.36, 5.89 16.25,6.57 14.97, 7.21 13.35, 7.72 11.59,7.70 10.13,7.53
19.58, 3.53 19.04,4.31 18.28, 5.11 17.33, 5.86 16.20, 6.53 14.91, 7.14 13.30, 7.61 11.54, 7.54 10.09, 7.32
i9.62, 3.41 19.09,4.21 18.34,5.03 17.35,x79 16.18,6.39 14.93, 6.90 13.42, 7.38 11.61, 7.33 10.24,6.99
19.66, 3.43 19.16,4.33 18.38, 5.28 17.28, 6.14 16.00, 6.74 14.76, 7.18 13.35, 7.82 11.32, 7.91 10.00, 7.51
8.44,6.78 7.35, 6.19 6.49, 5.73 5.04,5.21
4.76,6.59 3.84, 5.78 3.09, 5.10 2.23, 4.25
8.44, 7.04 7.34,6.51
8.44, 6.78 7.35, 6.19
8.58,6.49 7.58,5.86
8.15, 7.17 7.19, 6.66
6.44,
6.14
6.49,
6.77,
6.31,
6.21
4.57,
5.63
5.03, 5.22
5.36,4.82
4.64,
5.69
4.85, 3.76 3.39, 2.89 4.03, 2.58
2.18, 3.44 1.59, 251 1.58, 2.45
4.78,4.26 3.25, 3.26 3.89, 3.32
4.84, 3.76
5.07, 3.25
4.53,4.14
3.38, 2.50 4.01, 2.60
3.63, 2.51 4.22. 1.95
3.02, 3.21 3.74,3.15
Eq. (2)
a.0 .8.1 6.2 8.3 8.4 8.5 8.5 8.7 8.8
19.58, 3.53 19.(14,4.31 18.28,5.11 17.33, 5.86 16.20, 6.53 14.91,7.13 13.30, 7.61 11.54, 7.54 10.09, 7.31
18.69, 17.74, 16.48, 14.95, 13.24, 11.37, 9.37, 7.55, 6.09,
8.9 9.0 9.1 9.2 9.3 9.4 9.5 a)
LETTERS
Eq. (3)
Exact solution
, IogCflW
determined
PHYSICS
TRl & for the use using the exact solution ple TDS cluve.
with the time reference
5.73
point displaced
b) TIP2 has the time reference point displaced one data point back (-5
on datz
point
5.32
forward
(+5 ps) on the sam-
ps).
Table 2 Dielectric relaxation parameters for n-propanol derermined from different analyses of one set of time-domain spectroscopic dab Method
logcfo/Hz)
r/ps
EGO3) em b) E,,
8.68 8.5 8.65
332 500 350
3.7 1 3.2
3.4 2.9
20.6 20.5 20.6
a.58
332
3.7
3.4
20.6
8.58 8.63
420 370
3.5 3.5
; .
20.5 22.5
8.66
348
2.5
.
20.7
F-trnnsform
exact solution =q. (2) eq. (3) =q. (4) Fit
3. Complex plane d&am for n-propan deduced from the &ta of fig. 2. Points indiczted as q and 6 refer to timeref&nces bsFed on “steepest-slops” and “extipolntion to +=&x” resprctiely and differ by one data FOint (5 ps). Points indicated as + correspond to 2 shift of time-reference by one data point from e, giving tile sequence 8, Q, q .
Table 2 summarizes the relaxation parameters froth the different frequency domain methods, *
note
that
the values
obtained
using
eqs.
(I)
obtained and and (4)
.are.in.good ageement. Also shown in iable 2 are the parameters deduced using the time-do&n relations eqs. (6) and (7), and we see that the values obtained using the Cole method, eq. (7):, with the time reference advanced 27.ps, &e in good agreement with the best Four&r trgnsforrr, methods. Van Gemert (ref. [I],
time domain eq. (6) eq. (7) c) eq. (7) d) a)
Time referencing steepest slopes of transients without extiapolation of linear portions. b, Time rererencbg steepest slopes of transients with extrapolation of line= portions. Cl Time zero defined at steepest part of transient. 1, Time zero advanced 5 data points (27 ps) closer to peak of S-function
299 p. 545) K byquotes TDR. EO = 20.6,
a
E,
= 4.3 and TO = 325 PS at
Fig. 4, a plot of Iw![E*(w)] 1/2/c\ against logf, iS useful guide to which approximations are applicable.
:
.--::.“,i
-&: . . .,
_.y,-’
“.
‘_.,
..
:
:;. :.:
-.
:.
..
Volume 34, number
1 August 197.5
CHEhILCAL PHYSICS LETTERS
3
domain analysis using eq. (7) yield results in very good agreement with those obtained from solution of the “exact” relation, eq. (1). In view of the variation of the results with changes irr the method nf time-referencing (see fig. 2) the E’ and E” values at higher frequencies in the present range must be viewed with some caution.
The authors wish to thank the Science Research Council for support of this work, &chard Tuxworth for advice and provision of some of the Fourier transform programmes, Professor Cole, Dr. Claasen, Dr. Van Cemert and Dr. Suggett for preprints of their l.&
F/Hz
10
Fig. 4. plot of Iwl[o* (w)] ‘~1~1 (= 2dfXB, where XE is the wavelength inside the dielectric m&urn) against log f. Hori2ontaJ lines represent the upper limits of applicability of the vtious equations and the curres zre for n-propnnol in ozlls of different lengths. Lines A, B, C are at the upper limits of applicability ofeqs. (2), (3), (4) resPectiely. Line D is at the pmctical upper limit of eq. 111 znd line E is in the multiple solution region of eq. (l!.
work.
References [l] M.J.C. van Gemert, pzlilips Res. Rept. 28 (1973) 530. [2] H. Fellner-FeldeSg, J. Fnys. Chem. 76 (1972) 2116. [3] h5.J.C. vzn Gemert, J. Chem. Phys. 60 (1974) 3963. [4]
[5] [6]
Curves are drawn for n-propanol in cells of different lengths and a particular equation can only be used when the I/-~~[E*(w)] 1/2/c1 parameter is below the line representing the upper Limit of applicability for that particular equation. We therefore conclude that if the data are processed within the ranges of approximations indicated that the Fourier transform analysis using eq. (4) or the time
[7] [S] [9] [ 10)
A.H. Cl&,
P.A. Quickenden
and A. Susgctt, J. Chem.
Sot Faraday II 70 (1974) 1847. R.H. Cole, J. Phys. C&em. 78 (1974) 1440. M.J.C. van Gemert and W.H. de Jeu, Chenr. Phys. Letters 29 (1974) 287. h1J.C. van Gemert and A. Suggett, I. Chem. Whys., to be published. T.A.C.M. Clzxen and hfJ.C. van Gemert, J. Chem. whys., to be published. R.H. Cole, J. Phys. Chem. 79 (1975) 93. H.W. Loeb. G.M. Young, P.A. Quickenden and A. Suggett, Ber. Bunsenges. Physik. Chem. 7.5 (1971)
1155. [ll]
R.W. Tuxworth,
Ph.D. Thesis, Wales (1975).
CHEMICAL PHYSICS LETTERS
1 August 1975
PRACTICAL ASPECTS OF TIME DOhMALNREFLECTOlWETRY T.S. CrLAEKSON and G. WILLIAMS Edward Davies Ckmical Laboratories, Abayshvyth, SY23 IN!?, UK
The Univewity
College
of Wales,
Received 23 April 1975
Dielectric relaxation data For n-propanol ze obtained in the timedomtin and in the frequencydomain (vi;l Fourier transformation) using the method of time-domain refledtometry. The frequencydependent dielectric pennittivity was dedated nrrmerially using an exact solution 2nd various approximate methods. The accuracy OF the avzililable procaiures is assessed in the frequency
and time dam-As.
In
(wr/c)w(W)I
recent years there has been an interest in studyrelaxation behaviour using time-domain spectroscopy [l-9]. Of the various methods available that involve the measurement of the scattering coefficient S, (w) of a thin sample contained in a coaxial line terminated by its characteristic impedance, this is of particular interest. The prefent communication comments on the experimental method and on the
ss(w)‘&!
choice
Cole [9]
;
< 1.
(3)
ing dielectric
of method
for treating
the transient
data.
For the system described above, SQ(w) is given, in the usual notation [I], as
s,(o!=P(w)
1 - exp I-(Zwl/c)[E*(U)l 1 - $(w)exp
C-(2iwllc)
procedure.
There are now sev-
eral approximations of eq. (1) for which e*(w) can be solved analytically. Fellner-Feldeg [2] (thin cell method) S, (w) = (iwl/Zc)[l (2c&/c)l
- c*(w)] ,
[E*(W)]
r/21 4 1 .
van Gemert and de Jeu [6] kd
Sri(U)===
1 -E*(o)
1 +iw[l
+E*(0)]/2c’
&(cd/c)4lef(w)P 4 1. s&4 E3(Ccr)-1=21C wll
x =
[E* (o)ll~z~
(1) or graphical
1 -E”(O) fl 1 +(iwl/2c)[l +c*(w)] +$(iw1/2c)%“(w)’
‘So(w)
(4)
(x cothx f i&/c),
(5)
1121
S,(w) is determined experimentally and the complex permittivity E*(W) may be obtained from eq. (1) by a numerical
Claasen and van Gemert [g]
(2)
iwl[E*(ti)]
l/*/c .
Eq. (5) m.ay be expanded (via 3 series expansion of x cothx) which yields eqs. (3) and (4) as successive appro-ximations. ‘The experimental method we have wed was very similar to that of SugSatt and co-workers [4], and employed a Hewlett-Packard 18 1 TDR system (28 ps tunnel diode) used in conjunction with a PDP-12 computer for signal averaging and data-logging. The sampie length was typicalIy 5 mm contained between PTFE beads (see ref. [2]) which had been specially constructed
with respect
to the inner conductor
so
that they did not give rise to spurious reflections. As in ref. [4] two tiansients were measured, one with and-one without the sample in .tbe Line. The reflected signal from the sample was obtained as the difference .-
46!
Volume 34, number 3
CHEMICAL PHYSICS LETTERS
between the two traces [4]. For each trace 1 OGO digital points were recorded. Dielectric parameters could be determined in the time domain [S] or in the frequency domain by means of Fourier transformation. The transformation was carried out using the Samulon meth.od [lo] on an ICL 4130 computer and using programmes developed by Tuxworth [I I]- One important irregularity arising in the data was associated with spurious reflections at the sampler gate. By choosing an optimum arrangement of delay lines between the tunnel diode and the sampler, and the sampler and the sample, only the reflection samplersample-sampler-sample-sampler was evident and it was usually of srnaJ magnitude. There were also slight mis-matches at the E’TFE beaas. One important quantity is the base-line of the transient, and in the present work this was determin:d as an average of 30 datapoints starting at about 100 data-points before the reference step signnal. It was important that the records of time-domain data were accurately matched before subtraction and that the reflected waveform was accurately referred to the same time-origin as the incident pulse. In the present wclrk no external time-marker was used, so in order to identify a point on a record for matching with a corresponding point on another record two methods were used: (a) the point at which the slope of the transient is greatest is identified as a time-reference point, (b) a Ihear extrapolation from the steepest part of a transient to the voltage b=e-line [IO] defies a time-reference point. Referring to fig. 5b of ref. [4], using either referencing method, record C is matched to record D, then after subtraction but before Fourier transformation record C is matched to record E. For Fourier transformation a zero-time point is defmed on C and on E as corresponding points about 100 data-points bsfore the time-reference point. When analysis is made in the time-domain (vide infra) it becomes necessary to define a zero of time on the response record E. If the incident voltage pulse was a perfect step the zero would be at the first data point of curve E above the baseline. Howeve;, for a non-ideal step this zero may need. to be advanced by as much as the rise-time of the pulse:. The difference in timereferencing using methods (a) and (b) was found typically to be one data-point, i.e., = 5 ps. We have made measurements on n-propanol at 293 K ip a cell‘of liquid depth 5 mm and the following
cogsider the-analysisof the transient data.
paragraphi ,‘.
,-.
.’
197.5
For analysis
in the frequency domain the parameter lo![ea(w)]1/2/cl varied from 0.015 atflo’.5 Hz to 2.0 atf= 1O1o-o Hz. If we require that the criteria for the approximations to hold to be such thatx -=@1 should be taken to be x < 0.01 then in eqs. (2), (3) and (4) above we would require Iwf [E*(W)] l/2/ c I to be < 0.005, 0.17 and 0.82 respectively. This means that the latter two approximations would be useful for this system, up to 0.5 GFlz and 3 GHz. respectively. In practice the best procedure is to solve eq. (1) exactly, and the NewtonRaphson iteration method was used in the present work, and was found to be accurate and efficient under conditions such that Iw![E*(w)] lj2/,l < 1.5. However, at higher frequencies or with longer lengths I of dielectric two undesirable effects become apparent. First the fact that eq. (I) may lead to multiple solutions for E* for a given Sl, (w) may mean that the iteration procedure evaluates the incorrect solution to eq. (1). For any dielectric when IcJ[E*(w)] li2/cl e 2.2 there exists a pair of solutions of similar magnitude for E’ for a given S, (w). Second, we have observed that although the experimental data for the range 1.5 < Iwl[e*(w)] I/*/,! < 2.2 appeared quite accurate, reliable values for E*(W) could not be ob-
tained because of an ill-favoured region in the solution of eq. (1) - small changes in S, (0) lead to large changes in E”(O). Figs. la and lb illustrate this particular point. In fig. la reasonably accurate experimental values of the components of 27,(0) give correspondingly acceptable values for the components of P(w). In fig. lb, one may be in an ill-favoured region, and the values oft*, thus determined, could be subject to large errors. We conclude that the solution of the full equation, eq. (I), has the limitation although in theoIwl[e*(w)]l/2/cl < 1.5, in practice, ry there is no such limitation. Thus the extra frequency range or cell length that can be handied by the solution of eq. (l), via the Newton-Raphson or some similar iteration method, is only better than the best approximation [eq. (4)J by about a factor of 2. The
experimenter should plan applications of the present method to I~J[e*(w)]~‘~/cl < 1 S, using eq. (l), or if the criteria for the approximations [eqs. (2)-(4)] are met and computing facilities are more limited, the approximations can be used up to the values of IwI[e*(w)] ‘/!/cl indicated. The major part of t!e
computation will be the numerical Fourier Transformation.
452.’ :
1 Ausst
Volume
34, number
CHEMICAL
3
PHYSICS
1 August
LETTERS
1975
Fk,. 2. 5 ns transient
records for the line terminated by its impedance and containing a 5 mm cell. Tran-
chaacteristic sient
(a) empty
cd,
(b) cdl filled
with n-propanol
at 293 K.
Fellner-Feldegg [2,3] and Cole [S]. In the usual notation [2,3,5] : Fellner-Fe!degg thin.cell method [2,3] r(& r) = -(U2cro)(~o-~,) for
14
Cole
exp(-I/To), t/To
and
2CT&,-3Eml
4
(6)
~CT&O--Em).
method [5]
$(t) f (e--l)
= -(2c/l)
1 {r(t’)/Vo)dr’ 0
6) Fig. 1. Plots ofSU(w) on the complex plane E*(W). Continuous curves repreent Re S2(w) and the orthogonal set of broken curves reprexnts Im 31 (w). (nj w i/c = 0.1, (b) wl/c = 0.8, where there is a possibility of two E* vtilues for a particular S, value. Graphs such as these can be used to solve eq. (1).
A low frequency limit for the Fourier transform aisses from uncertainties in methods [eqs. (l)-(4)] the base-line determination, owing, in part, to droop on the base-line. In our experiments errors were noticeable at frequencies below 107e5 Hz, and positive values of ReS,, (0) tid hence negative values of E” may be obtained. The measurements using a 5 mm cell for n-propanol
together
with eqs. (l)-(4)
may
therefore give acceptable results in the frequency range 107m5to 10gm5Hz. Two approximations are available for the analysis of transient data in the time domain and are due to
i (2c/l) j
(r(t') r(r'-f)/Vg)
dt’-ti(t)jVo
.
(7)
0
We have used eqs. (l)-(4) and eqs. (6) and (7) for an analysis of transient data for n-propznol in a 5 mm cell at 293 K. Fig. 2 shows a 5 ns scan for the transients for the empty and k-tied cell. Table 1 lists E’and E” obtained over the frequency range using the different
methods
of analysis
in the frequency
don&u
the steepest-slope time referencing method. The results from eqs. (2). (3) and (4) all agree with those obtained by solution of eq. (!) at low frequencies, and we see that the results using eq. (4) compare very satisfactorily with those from eq. (1) at all frequencies, in confirmation of Claascn uld van Gemert [8]. Changing the time-reference by one datz-point (= 5 ps) has a large effect on the values for E’and E”
given
c&dated
from the exact
relation,
eq. (1). This is
shown in table I and in fig. 3 - the Argand
dizg-ram.
463
Volti~&
W, number
3
CHEMICAL
Table 1 Values of (E’, l n) at different
frequencies
1 August 1975
by eqs. (l)-(4) Eq. (4)
TRL a)
TR2 b)
5.34 6.35 7.30 8.07 8.60 8.89 8.81 8.21 7.53
19.60, 3.53 19.06,4.32 18.31,5.13 17.36, 5.89 16.25,6.57 14.97, 7.21 13.35, 7.72 11.59,7.70 10.13,7.53
19.58, 3.53 19.04,4.31 18.28, 5.11 17.33, 5.86 16.20, 6.53 14.91, 7.14 13.30, 7.61 11.54, 7.54 10.09, 7.32
i9.62, 3.41 19.09,4.21 18.34,5.03 17.35,x79 16.18,6.39 14.93, 6.90 13.42, 7.38 11.61, 7.33 10.24,6.99
19.66, 3.43 19.16,4.33 18.38, 5.28 17.28, 6.14 16.00, 6.74 14.76, 7.18 13.35, 7.82 11.32, 7.91 10.00, 7.51
8.44,6.78 7.35, 6.19 6.49, 5.73 5.04,5.21
4.76,6.59 3.84, 5.78 3.09, 5.10 2.23, 4.25
8.44, 7.04 7.34,6.51
8.44, 6.78 7.35, 6.19
8.58,6.49 7.58,5.86
8.15, 7.17 7.19, 6.66
6.44,
6.14
6.49,
6.77,
6.31,
6.21
4.57,
5.63
5.03, 5.22
5.36,4.82
4.64,
5.69
4.85, 3.76 3.39, 2.89 4.03, 2.58
2.18, 3.44 1.59, 251 1.58, 2.45
4.78,4.26 3.25, 3.26 3.89, 3.32
4.84, 3.76
5.07, 3.25
4.53,4.14
3.38, 2.50 4.01, 2.60
3.63, 2.51 4.22. 1.95
3.02, 3.21 3.74,3.15
Eq. (2)
a.0 .8.1 6.2 8.3 8.4 8.5 8.5 8.7 8.8
19.58, 3.53 19.(14,4.31 18.28,5.11 17.33, 5.86 16.20, 6.53 14.91,7.13 13.30, 7.61 11.54, 7.54 10.09, 7.31
18.69, 17.74, 16.48, 14.95, 13.24, 11.37, 9.37, 7.55, 6.09,
8.9 9.0 9.1 9.2 9.3 9.4 9.5 a)
LETTERS
Eq. (3)
Exact solution
, IogCflW
determined
PHYSICS
TRl & for the use using the exact solution ple TDS cluve.
with the time reference
5.73
point displaced
b) TIP2 has the time reference point displaced one data point back (-5
on datz
point
5.32
forward
(+5 ps) on the sam-
ps).
Table 2 Dielectric relaxation parameters for n-propanol derermined from different analyses of one set of time-domain spectroscopic dab Method
logcfo/Hz)
r/ps
EGO3) em b) E,,
8.68 8.5 8.65
332 500 350
3.7 1 3.2
3.4 2.9
20.6 20.5 20.6
a.58
332
3.7
3.4
20.6
8.58 8.63
420 370
3.5 3.5
; .
20.5 22.5
8.66
348
2.5
.
20.7
F-trnnsform
exact solution =q. (2) eq. (3) =q. (4) Fit
3. Complex plane d&am for n-propan deduced from the &ta of fig. 2. Points indiczted as q and 6 refer to timeref&nces bsFed on “steepest-slops” and “extipolntion to +=&x” resprctiely and differ by one data FOint (5 ps). Points indicated as + correspond to 2 shift of time-reference by one data point from e, giving tile sequence 8, Q, q .
Table 2 summarizes the relaxation parameters froth the different frequency domain methods, *
note
that
the values
obtained
using
eqs.
(I)
obtained and and (4)
.are.in.good ageement. Also shown in iable 2 are the parameters deduced using the time-do&n relations eqs. (6) and (7), and we see that the values obtained using the Cole method, eq. (7):, with the time reference advanced 27.ps, &e in good agreement with the best Four&r trgnsforrr, methods. Van Gemert (ref. [I],
time domain eq. (6) eq. (7) c) eq. (7) d) a)
Time referencing steepest slopes of transients without extiapolation of linear portions. b, Time rererencbg steepest slopes of transients with extrapolation of line= portions. Cl Time zero defined at steepest part of transient. 1, Time zero advanced 5 data points (27 ps) closer to peak of S-function
299 p. 545) K byquotes TDR. EO = 20.6,
a
E,
= 4.3 and TO = 325 PS at
Fig. 4, a plot of Iw![E*(w)] 1/2/c\ against logf, iS useful guide to which approximations are applicable.
:
.--::.“,i
-&: . . .,
_.y,-’
“.
‘_.,
..
:
:;. :.:
-.
:.
..
Volume 34, number
1 August 197.5
CHEhILCAL PHYSICS LETTERS
3
domain analysis using eq. (7) yield results in very good agreement with those obtained from solution of the “exact” relation, eq. (1). In view of the variation of the results with changes irr the method nf time-referencing (see fig. 2) the E’ and E” values at higher frequencies in the present range must be viewed with some caution.
The authors wish to thank the Science Research Council for support of this work, &chard Tuxworth for advice and provision of some of the Fourier transform programmes, Professor Cole, Dr. Claasen, Dr. Van Cemert and Dr. Suggett for preprints of their l.&
F/Hz
10
Fig. 4. plot of Iwl[o* (w)] ‘~1~1 (= 2dfXB, where XE is the wavelength inside the dielectric m&urn) against log f. Hori2ontaJ lines represent the upper limits of applicability of the vtious equations and the curres zre for n-propnnol in ozlls of different lengths. Lines A, B, C are at the upper limits of applicability ofeqs. (2), (3), (4) resPectiely. Line D is at the pmctical upper limit of eq. 111 znd line E is in the multiple solution region of eq. (l!.
work.
References [l] M.J.C. van Gemert, pzlilips Res. Rept. 28 (1973) 530. [2] H. Fellner-FeldeSg, J. Fnys. Chem. 76 (1972) 2116. [3] h5.J.C. vzn Gemert, J. Chem. Phys. 60 (1974) 3963. [4]
[5] [6]
Curves are drawn for n-propanol in cells of different lengths and a particular equation can only be used when the I/-~~[E*(w)] 1/2/c1 parameter is below the line representing the upper Limit of applicability for that particular equation. We therefore conclude that if the data are processed within the ranges of approximations indicated that the Fourier transform analysis using eq. (4) or the time
[7] [S] [9] [ 10)
A.H. Cl&,
P.A. Quickenden
and A. Susgctt, J. Chem.
Sot Faraday II 70 (1974) 1847. R.H. Cole, J. Phys. C&em. 78 (1974) 1440. M.J.C. van Gemert and W.H. de Jeu, Chenr. Phys. Letters 29 (1974) 287. h1J.C. van Gemert and A. Suggett, I. Chem. Whys., to be published. T.A.C.M. Clzxen and hfJ.C. van Gemert, J. Chem. whys., to be published. R.H. Cole, J. Phys. Chem. 79 (1975) 93. H.W. Loeb. G.M. Young, P.A. Quickenden and A. Suggett, Ber. Bunsenges. Physik. Chem. 7.5 (1971)
1155. [ll]
R.W. Tuxworth,
Ph.D. Thesis, Wales (1975).