The resolution function

In this Appendix we derive the analytical expression for the energy resolution of a low-bandpass spectrometer like TOSCA ( 3.1) (also known as crystal analyser spectrometers) and describe two key features of the design ( 3.2), time focussing ( 3.2.1) and the Marx principle ( 3.2.2) that improve the resolution at high and low energy transfer respectively. 

The resolution function for crystal analyser instruments is complex but can be summarised as [1,2]  

In this case F = (FWHM)/2.35. (For components that are better approximated with a boxcar distribution, e.g. detector and sample width, r=(FWHM)/(2 /3).) 

The quadratic terms will be dealt with in the order they appear in Eq. (A3.2). 

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In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

Figure 6.3 Schematic of the resolution function in the reciprocal space. Two vectors ki and kd represent the incident and the diffracted wave vectors and H represents the scattering vector. The divergence of the incident X-ray and the acceptance window of the diffracted beam side optic are represented as Suj and 6(29).

The resolution function and short lifetimes remain rather constant throughout the porosity range. The longer lifetimes increase steadily with porosity up to about 50% porogen load. At larger loads the lifetimes remain similar while their intensities drop. 

Subtraction of the resolution function reveals the vibrational contribution S (v) to the excitation probability. PHOENIX adjusts the subtraction weight to achieve the best match to the one-phonon contribution to the vibrational signal near Eq expected for a Debye frequency distribution (D (v) a v y The Fourier-log algorithm then yields the dominant first-order vibrational contribution... 

The approximation of the resolution function components as Gaussian or Lorentzian functions is an approximation. Furthermore incorporation of the resolution function as a convolution in t is an approximation. 

The resolution function for these components is represented as a Gaussian in t, with standard deviation A/., //. 

Again there is a slight increase in the anomalies when the DD method is used. This is the opposite trend to that which would be expected, if the anomalies were due to the method used to incorporate the resolution function. 

Fig. 3.21(a) The resolution function and its components for TFXA, (b) the evolution of the resolution on TFXA, TOSCA-1 and TOSCA. 

It is possible to obtain a limited improvement in resolution by reducing the thickness of the sample, graphite and analyser as seen for TOSCA 1 relative to TOSCA. If the these are reduced too much then the other terms in the resolution function will dominate and no further improvement will be seen. The price of reducing these terms is that the detected flux decreases, so compensating action is also required here. 

Eq. (A3.30) gives a reasonable description of the resolution function and qualitatively reproduces the behaviour well. In detail, it is found to underestimate the resolution observed experimentally. The problem stems from Eq. (A3.21) which predicts a divergence in the analyser that is larger than observed. The reason for this is that all the variables are assumed to be independent and this is only an approximation. Recent work on the development of a TOSCA-like spectrometer on the SNS source VISION (Oak Ridge, USA) has lead to a significant improvement in the mathematical representation of the resolution elements of these types of spectrometer. At the time of going to press this work was unpublished [4]. 

Two typical spectra obtained with wet and dry samples are shown in fig. 13. For the wet sample, the NQES spectra are composed of a sharp peak reproducing the resolution function of the instrument superimposed on a broad component having about 100y eV FWHM. The dry sample on the contrary, exhibits only the sharp peak. This observation immediately tells us that the broad... 

Experimental data (Fig. 43a) indeed are fit by the structure factor of Leibler s theory [43] nicely - the curves shown in this figure from a convolution of Eq. (187) with the resolution function, as shown in the insert (the dashed curve is Eq. (187), the full curve is result of the convolution, for T = 126.3 °C). However, at each temperature both Rg(T) and x(T) are used as adjustable parameters - therefore a nearly perfect fit is possible although the peaks at the different temperatures do not occur at precisely the same q. Thus the agreement shown in Eq. (187) should not be taken as a proof for the accuracy of Leibler s theory -such a proof would require an independent measurement of Rg(T) and y(T). 

The general aim of chromatographic separation is the segregation of individual compounds from a sample mixture, as in Fig. 2-1. The relative success of this operation in the case of two compounds that migrate adjacently through the adsorbent bed is determined by the relative overlap of the two bands at the end of separation. Figure 2-6 portrays this situation for an adjacent pair of adsorbed bands. The resolution of the two bands is determined by two factors the relative separation of the band centers (d — d ) and the widths of the individual bands (cTj and a ). We will define the resolution function equal to (d — d f2 a -f ffj). 

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