Informe de Evaluación Mutua contra el Lavado de Dinero y Contra el Financiamiento del

In principle, the sum in equation (1) should run over all (21 + 1) components o f a mode but some components may overlap (geometric deformations o f the cavity were small and, it would seem, approximately axisymmetric) or they m ight be weakly excited or detected because o f the positions o f the microwave probes. In any case, even for the TM1« modes, with only 3 components, the number o f adjustable parameters in equation (1) could be as large as 15 i f the sum ran over all components. O f the five modes measured, it was only for the three-component TM1« modes that there was a real possibility o f satisfactorily resolving the components and, consequently, obtaining reliable estimates o f the component halfwidths gpf. The observed values o f quality factor for individual components,

Q n - / n varied from about 14000 for T M l 1 at 360 K to 37400 fo r T M l3 at

189 K. It is d ifficu lt to maintain, with confidence, that individual components o f the TM21 and TM31 modes were properly resolved, but, i f we were, indeed, successful in doing so, then it appears that the quality factors o f individual components varied from about 10500 at 360 K to 14250 at 189 K for TM31, and from about 18250 at 360 K to 22600 at 189 K for TM21.

Superficially, the resonances o f the sphere appeared to consist o f either one component (‘singlets’) or a pair o f components (‘doublets’), as can be seen in figure (6.9) where the responses o f the evacuated sphere near 189 and 360 K are shown; the measurements were taken at the end o f the respective isotherms in xenon. The

appearance o f the responses was very sim ilar in the evacuated and the gas-filled cavity at each temperature, and was reasonably consistent w ith an approximately axisymmetric deformation o f the spherical resonator [see section (4.6)]. Therefore, modes T M l2 and T M l 3, which ought to consist o f three components, were always analysed as singlets, w hilst modes T M ll, TM21 and TM31, which ought to consist o f three, five and seven components, respectively, were always analysed as doublets. This approach invariably led to convergent and stable fits o f the transmitted powers to equation (1), which was never found to be the case when triplet fits were used; higher orders o f fit, for modes TM21 and TM31, were never attempted. The number o f parameters in the singlet and doublet fits was chosen on the basis o f whether there was a statistically-significant reduction, at the 0.995 probability level, in the overall standard deviation o f the fit, and a similar reduction in the reduced chi-squared

statistic on addition o f extra (background) parameters. In the context o f this

work, was defined as

where was the difference between the measured power and the

power calculated using equation (1) at drive frequency / M was the number o f data points (always 31 in this work), n was the number o f parameters in the fit, and

(^(APY) was the average squared difference between the up and down-sweep powers

at each point, which was used as an estimate o f the variance o f the data [see section (6.4)]. W ith an appropriate number o f parameters, it was usually possible to fit

equation (1) to the data for any o f the five modes w ith Xt less than unity (the ideal

value). That x f was generally much smaller than unity implies that the estimated

variance [(APY ) provided a rather pessimistic account o f the collected data. It was

also necessary for any extra parameters to be significantly different from zero for their inclusion in the fit, and this was assessed by comparing the magnitude o f the

ratio (z/cr^), where Z is a parameter value and is its standard deviation, w ith the

Figure (6.9) Measured responses o f the evacuated sphere at 188.5533 and 360.2599K T Ml 1 E CL CL 1 8 8 .5 5 3 3 K 3 6 0 .2 5 9 9 K - 0 . 3 - 0 . 2 -0 .1 0.0 0.1 T Ml 2 E CL CL 1 .0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 8 8 .5 5 3 3 K 36 0 .2 5 9 9 K - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 -0 .1 0.0 (/^ —/’i2)/MHZ 0.1 0.2 E CL TMl 3 1 8 8 .5 5 3 3 K 3 6 0 .2 5 9 9 K 0.9 0.8 0.7 0.6 0.5 0 . 4 0.3 0.2 - 0.3 - 0.2 - 0.1 0.0 0.1 0.2 0.3 0.4 0.5 197

T M 2 1 0.9 0.8 0.7

I

The singlet fits, for modes T M l2 and T M l3, could contain three, five or seven

(adjustable) parameters: and g]\[ for the three-parameter fits without

background, \a^ \ , f y , g]\f, | | and arg(jB) for the five-parameter fits with a constant background, and U jv L fN^ gN^ I jB I , arg(5), | c | and arg(C) for the seven-parameter fits with a combination o f constant and frequency-dependent background. For all the T M l3 mode measurements taken in this work, the five parameters \Aj\[\, f y , gj\[, 1^1 and arg(jB) were always required to satisfactorily accommodate the data, but the extra background terms | C | and arg(Q were rarely deemed to be significant. To promote consistency along individual isotherms and between the measurements at different temperatures, the five-parameter fits were always used. With five adjustable parameters, equation (1) could be generally made

to fit T M l3 mode data with a fractional standard deviation [o{P)IP^^ o f about

0.5 %, where o{P) was the standard deviation o f the power obtained in the regression with equation (1) and P^^^ was the maximum transmitted power. As w ill be seen in

chapter 7, it seems that only one o f the three T M l3 mode components was predominant in the experimental responses, w ith the remaining two (weaker) components at higher frequency than the quarter-power-point range over which measurements were taken. This accounts for the success o f the singlet fits, w ith the low-frequency ‘ta il’ o f the weaker components being accounted fo r by the background terms. By analogy w ith calculations given in reference 132, an estimate o f the fractional random uncertainty in the halfw idth o f a component is given by

[ ^ nVs n] == 2[o(f)/fm ax], an estimate o f the fractional random uncertainty in

the resonance frequency o f a component is given by [o(/Xr)//5v] = [o(g]\f)/f]sf\ =

[oi^yPjnaxVQN' W ith quality factors o f between about 30300 at 360 K and 37400 at

189 K, [o(/jv)//5v] was generally less than 0.2 ppm for the T M l3 mode. Such

estimates o f [oigNVgN] and were always at least as large as the fractional

standard deviations o f repeated measurements o f gj\/ and fy/, for any mode o f the sphere or cylinder, w hilst the temperature and pressure were kept constant.

For the T M l2 mode, the number o f significant parameters was generally more

ambiguous. For the measurements on nitrogen, xenon and the mixture

{0.5 A r + 0.5 N j} at 300 K and argon at 215, 260 and 300 K, the fu ll seven

parameters gjy, 1^1, arg(5),

|c|

significant, whereas for the remaining measurements on xenon at temperatures from 189 to 360 K, only five parameters were required, although there was often some ambiguity in choosing between the five and seven-parameter fits. The reason for such changes in the order o f fit is not entirely clear, but it was necessary to renew the indium seal on one o f the microwave probe mounts, before the measurements on xenon between 189 and 360 K (apart from that at 300 K ) could be taken, and its replacement presumably gave rise to a subtly different background, rendering the inclusion o f the | C | and arg(C) parameters unnecessary. In any case, provided the same order o f fit was used for all points along an isotherm, the differences between the estimates o f refractive index obtained from the five and seven-parameter fits was generally less than 1 ppm. Using five or seven adjustable parameters, as appropriate, it was generally possible to fit equation (1) to the T M l2 data with a fractional

standard deviation [o(P)/Pj„^] o f between 0.5 and 2.0 %. As w ill be further discussed in chapter 7, it appears that the experimental responses for mode T M l2