Because of the nonuniformity of the target observed in
Section 3.3, the average layer thickness depends on the
width and profile of the beam which stops in the target. Also recall that
we have measured only the profile in the Y (vertical) dimension, hence
the horizontal profile has to be assumed. The effective thickness can be
defined via
(81) 
(82) 
Figure 6.1 illustrates the dependence of
the effective thickness on the beam parameters. The average was calculated
assuming rotational symmetry of the thickness profile, and weighted with two
different beam parameterizations (i) Gaussian beam, and (ii)
a flat top Gaussian, for which the radial intensity at the
distance r, f(r) is defined by:
The figure also shows the dependence on the radial cutoff values which reflect the physical limit of the beam radius. For the upstream layer, the cutoff value can be considered to be 32.5 mm which is the radius of the thin gold target support frame (beyond this radius the muon would have to go through more than a millimeter of copper). As for the downstream layer, which is directly facing the upstream layer, there is no collimation of the beam due to the target support frame, hence an R cutoff of 35 mm in the X direction (from the external rectangular size of the gold plated copper frame) and slightly larger in the Y dimension can be expected. As can be seen from Fig. 6.1, the average thickness depends on beam profile, especially if the beam width is large.
Since the knowledge of the thickness is very important, all the available information needs to be combined to determine the effective thickness. We attempted this in the following manner: (1) first parameterize the muon beam distribution (in the XY plane) from the image of decay electrons obtained by the MWPC system, (2) then use that as an input to the Monte Carlo simulation to calculate the distribution of the beam reaching the downstream layer, and (3) finally take a weighted average for each of the upstream and downstream layers, using the assumed beam profiles. Note that depending on the thickness of the upstream moderator, the beam profile at the downstream layer, hence the effective thickness, could be different.
The knowledge of the beam profile is also necessary for the determination of the silicon detector acceptance, which is tabulated in Tables 6.16.3, together with the effective thicknesses, but will be discussed in the following section.
Figures 6.2 and 6.3 compare the experimental data and simulations, with different input parameters, of the Ydistribution of the decay electrons image in the upstream layer. The data were obtained via the MWPC imaging system, while the Monte Carlo code, SMC, simulated the imaging process in the detector with the initial muon beam stopping distribution and the wire chamber resolutions as input parameters. In Fig. 6.2, Gaussian distributions with varying FWHM were assumed for the initial beam distribution in the XYplane, while flattop Gaussian distributions, defined by Eq. 6.3, with varying R_{flat} and were used for Fig. 6.3. The Gaussian beam of FWHM mm, and the flattop Gaussian with the flat top radius R_{flat} of mm and Gaussian FWHM_{g}^{} of mm seem to reproduce the experimental data rather well. The resulting effective thicknesses are summarized in Table 6.1. The variation in the thicknesses with these values of beam parameters is less than 3%. The wire chamber resolution, , of 1 mm is used for this analysis, but variation of the wire chamber resolutions between 0.2 mm and 4 mm did not affect this conclusion.

The profile of the atomic beam reaching the downstream layer generally depends on, but differs from, that of muon beam stopping in the upstream layer. The SMC, with all the physics in it, was used to simulate the former, using the latter as SMC input. The resulting profiles were parameterized similarly to the upstream case.
Figure 6.4 illustrates an example of the simulated radial profiles of the beam reaching the downstream layer (plotted with error bars), for which the input to the simulation of a flattop Gaussian beam, with R_{flat}=12 mm and FWHM_{g}=12 mm, was assumed for the upstream beam profile. Shown as a histogram is a parameterization of that profile using a flattop Gaussian function. The flat radius of 4 mm and Gaussian FWHM of 28 mm give a reasonable per degree of freedom (DOF) of 1.05. If a Gaussian distribution is assumed, a FWHM of 32.4 mm with DOF of 1.80 was obtained in the fit.
Table 6.2 summarizes the effective thickness for the downstream layer with the different US beam distribution. For the US beam of 1010 mm, a single Gaussian function fits the simulated distribution entering the DS target reasonably well, but for larger US widths, the fitted FWHM values depend on the region of the fit (e.g. [29;29] mm vs. [35;35] mm), and a flattop Gaussian function gave better as well as a more stable fit. The different DS beam parameters for the same US parameter in the table reflect variations due to the fitting region size. The deuterium overlayer of 48.3 ( T) was assumed for the simulations.
As discussed in Section 3.3.2 in Chapter 3 (see Fig. 3.16 in page ), the layer thickness depends on the distance between the diffuser surface and the target foil surface according to our film deposition simulations. Although the distance for the upstream layer was similar to the calibration measurements in Section 3.3 ( mm), that for the downstream foil was closer, i.e. mm. Therefore the calibration factors have to scaled by a factor . This factor was determined by comparing the areas between the simulations for the foil distance 8 mm and 2.8 mm in Fig. 3.16. About 3% uncertainty in the scaling is due to the choice of the interval within which the ratio was taken; [15, 15] mm and [30, 30] mm are the two extreme intervals considered, and the average between them was used as the scaling factor.
In general, the beam profile at the downstream layer depends on the thickness of the overlayer in the upstream layer. This is partly due to the more forward peaked angular divergence of going through the overlayer, and also because the moderated with larger angles are less likely to survive to reach the downstream layer due to the longer flight path. This effect was investigated in Table 6.3 with the US beam of 1010 mm assumed, and found to have about 6% effect in the effective thickness.