Effective target thickness

Because of the non-uniformity 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

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:

**Figure 6.1:** Dependence of the effective thickness (above) and the effective deviation (below) on the beam parameters and the radial cut-off values. (a) (b) (c): Flat-top Gaussian beam (FG, defined in Eq. 6.3) with and FWHM . (d) (e) (f): Gaussian beam (G) with FWHM_g = $\alpha$ . The points are slightly shifted horizontally for visual ease. The error bars represent the 5% uncertainty from the stopping power.

The figure also shows the dependence on the radial cut-off values which reflect the physical limit of the beam radius. For the upstream layer, the cut-off 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 $\mu t$ beam due to the target support frame, hence an R cut-off 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.

**Figure 6.2:** The Y-distribution of decay electrons in the upstream layer imaged by the MWPC system plotted with error bars is compared with the Monte Carlo simulations, in the histograms, assuming Gaussian distributions of FWHM 20, 25, and 30 mm (from inside out) as the initial muon beam distribution in the XY plane.

**Figure 6.3:** Similar to Fig. 6.2 except simulations are with the initial muon distribution being Gaussian, with flat top. The histograms, from inside out, show the beam distribution of 88, 1010, 1212, 1515, where first number is the flat top radius **R_flat** and the second is the full width at half maximum of the Gaussian part FWHM (see Eq. 6.3).

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 $\mu t$ 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 $\mu t$ 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 6.1-6.3, 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 flat-top 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 flat-top 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.

Table 6.1: The effective upstream (US) layer thickness with different beam parameters. Also shown is the corresponding acceptance , for the silicon detector.

US beam		effective thickness
R_flat (mm)	FWHM_g (mm)	( $\mu$ gcm^-2 (T $\cdot l$ )^-1)	(%)
0	20	3.461	2.355
0	25	3.373	2.392
10	10	3.493	2.336
12	12	3.449	2.394
15	15	3.281	2.462

The profile of the $\mu t$ 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 $\mu t$ beam reaching the downstream layer (plotted with error bars), for which the input to the simulation of a flat-top 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 flat-top Gaussian function. The flat radius of 4 mm and Gaussian FWHM of 28 mm give a reasonable $\chi ^{2}$ 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 $\mu t$ 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 flat-top Gaussian function gave better $\chi ^{2}$ 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 $\cdot l$ ) 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.

Table 6.2: The effective thickness and the silicon detector acceptance ( ) for the downstream (DS) layer, evaluated with different parameterizations of the $\mu t$ beam profile at DS, which in turn were simulated using different US profiles as input to the Monte Carlo. The upstream moderating overlayer of 14 T $\cdot l$ is assumed for the simulation. See the text for the details. The summary of effective thicknesses and Si solid acceptance will be given in Table 6.5.

US beam (mm)		DS beam (mm)		Effective thickness
R_flat	FWHM_g	R_flat	FWHM_g	( $\mu$ gcm^-2 (T $\cdot l$ )^-1)	(%)
10	10	0	28.8	3.60	2.44
12	12	0	32.1	3.50	2.47
		0	33.8	3.45	2.48
		4	28	3.53	2.46
15	15	0	42.1	3.25	2.50
		0	37.2	3.36	2.53
		7.5	30	3.36	2.51
0	20	0	28.0	3.63	2.43
0	25	0	31.8	3.51	2.47

In general, the $\mu t$ 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 $\mu t$ going through the overlayer, and also because the moderated $\mu t$ 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.

Table 6.3: The effective thickness of the downstream (DS) layer with different upstream overlayer thickness, and the corresponding silicon detector acceptance. The US beam parameter of the flat-top Gaussian 1010 mm was used as input for simulations.

US overlayer	DS beam (mm)		Effective thickness
thickness (T $\cdot l$ )	R_flat	FWHM_g	( $\mu$ gcm^-2 (T $\cdot l$ )^-1)	(%)
0	0	35.8	3.40	2.49
3	0	32.1	3.50	2.47
6	0	30.6	3.55	2.46
14	0	28.7	3.60	2.44