MAD at 12°, 20°, and 35°

December 21, 2004

Here are summarized the properties of the MAD at 12°, 20°, and 35° as determined from raytracing studies.

Central Momentum	8 GeV/c	Bend Angle	30°	Target Length		≥ 1 m

Configuration		~ΔΩ (m-sr)[*]	vert/horiz	min δ	max δ	σδ	σθ	σy (m)	σφ
35°	"Normal"	24	3.5	-16%	42%	0.0012	0.0004	0.0042	0.0009
20°	"Normal"	14	3.25	-20%	40%	0.0010	0.0003	0.0055	0.0009
20°	no quads	3	2 [†]	-38%	38%	0.0006	0.0001	0.0029	0.0003
12°	"Normal"	6	2.8	-21%	42%	0.0009	0.0003	0.0068	0.0007
12°	no quads	2.4	1.4^†	-39%	42%	0.0006	0.0001	0.0043	0.0003

Table 1: Summary Table. The results of the simulations discussed and presented below are summarized for quick reference in the table above. Resolutions quoted are for 8 GeV/c central momentum, are averaged over the full momentum acceptance, and include effects from a 1 m long target. For variations within the momentum acceptance see the more detailed plots presented below.

Overview:

Using magnetic field maps[‡] SNAKE based models of the MAD at 12°, 20° and 35° were assembled. These models include reasonably realistic magnetic fields[§] and realistic apertures in the magnets. The models were used to generate large numbers (3000-4000) of trajectories that varied randomly across the acceptance. Using those trajectories transfer functions were developed that reconstructed δ, θ₀, y₀, and φ₀ from position and angle measurements in the detectors. Then the effects of detector errors were evaluated using a Monte-Carlo type analysis. Each of the 4 detector parameters (x, θ, y, and φ) were smeared using a Gaussian random number generator with σ’s determined from a simple-minded, back of the envelope estimate of detector performance (see below). Resolutions were determined by comparing the actual target parameters (δ, θ₀, y₀, and φ₀) to those obtained using the smeared detector parameters in the transfer functions (the “measured” target parameters)[**]. Each trajectory was evaluated 100 times and the resolutions reported are the root mean squared differences between the actual and “measured” target parameters for all the trajectories in a given momentum bin. Two different tunes were considered at both 12° and 20°: a normal tune with quadrupoles on and a just dipoles tune, in which case the quadrupoles are left off. Results are reported for:

Baseline: In this case no smearing was applied. This demonstrates how good the generated transfer functions were by themselves.
8 GeV/c: Here the detector errors were those determined from the back-of-the-envelope calculation for 8 GeV/c central momentum taking into account momentum variations from bin to bin.
4 GeV/c: Same as 8 GeV/c but for 4 GeV/c central momentum.
A simple generic set of detector parameters, σ_x = σ_y = 100 μm and σ_θ = σ_φ = 0.5 m-rad.
Also plotted are the results of the previous TRANSPORT analysis (using the generic errors) for comparison[††]. (see report)

At the bottom are tables giving the size of the beam envelope in the detector region. For the “no quadrupole” tunes the envelopes are given both for full acceptance (everything that gets through the magnets) and for a smaller acceptance driven by the design size of the calorimeter (3 x 1 m²) (see envelopes)

Quick links:

RE Momentum Acceptance: Note that in this analysis the momentum acceptance is considerably larger than in previous descriptions of the MAD. This is not the result of any fundamental design change but merely the result of the observation[‡‡] that more gets through the magnets than originally intended.

Figure 1: Expected momentum resolution for MAD at 12° in the “normal” tune (Quads on). Notice the extra momentum acceptance up to +40%

Figure 2: Same as Figure 1 for vertical angle resolution.

Figure 3: Same as Figure 1 for transverse target position.

Figure 4: same as Figure 1 for horizontal angle resolution

Figure 5 Relative acceptance of MAD at 12° in 10% bins in delta. Approximate acceptance at delta = 0 is ΔΩ~6 m-sr.

Figure 6: Momentum resolution in MAD at 12° with quads off. With quads off solid angle is reduced but momentum range is very large (see Figure 10).

Figure 7: Same as Figure 6 for vertical angle resolution.

Figure 8: Same as Figure 6 for transverse target position resolution.

Figure 9: Same as Figure 6 for horizontal angular resolution.

Figure 10: Relative acceptance for MAD at 12° with quads off in 10% momentum bins. Approximate solid angle acceptance at delta = 0 and constrained to fit into a 1 x 3 m² Calorimeter is ΔΩ~2.4 m-sr. (see below)

Figure 11: Expected resolutions for MAD at 20°. Resolutions for Momentum (σ_δ), Vertical angle (σ_θ), Transverse target position (σ_y), and horizontal angle (σ_φ) are shown.

Figure 12: Relative momentum acceptance for MAD at 20°. Approximate solid angle acceptance for a thin target at delta = 0 is ΔΩ~14 m-sr.

Figure 13: Expected resolutions for MAD at 20° with quadrupoles turned off. Resolutions for Momentum (σ_δ), Vertical angle (σ_θ), Transverse target position (σ_y), and horizontal angle (σ_φ) are shown.

Figure 14: Relative momentum acceptance for MAD at 20° with quadrupoles off. Approximate solid angle acceptance for a thin target at delta = 0 and constrained to fit into a 3 x 1 m² Calorimeter is ΔΩ~3 m-sr (see below).

Figure 15: Expected resolutions for MAD at 35° “normal tune”. Resolutions for Momentum (σ_δ), Vertical angle (σ_θ), Transverse target position (σ_y), and horizontal angle (σ_φ) are shown.

Figure 16: Relative momentum acceptance for MAD at 35° “normal tune”. Approximate solid angle acceptance for a thin target at delta = 0 is ΔΩ~24 m-sr

Multiple Scattering Evaluation of Detectors:

In the so-called back of the envelope evaluation, multiple scattering[§§] is considered from several sources:

The Helium bag (15.3 m of He at 1 Atm.)
The Helium bag exit window (1-mil Kapton)
Air between the exit window and the 1^st chamber (0.3 m)
The 1^st chamber (0.10% RL [***])
Air between the 1^st and 2^nd chambers (1.0 m)

The effect of each source on position and angle determination at the detector package is evaluated separately and then all sources are added in quadrature. Also included are the intrinsic position and angle resolution of the chambers themselves (80 μm and 0.11 m-rad‡). The effects are evaluated separately for reach momentum bin considered. The table below shows a summary for 8 GeV/c electrons.

m	0.511 MeV
p	8000 MeV/c
z	1

	RL	θ₀	length	σy	σθ
Helium bag	0.270%	0.000068	15.3 m	0.00060 m	3.95E-05
Bag exit window	0.009%	0.000010	0.3 m	0.00000 m	5.97E-06
Air between exit and chambers	0.099%	0.000039	0.3 m	0.00001 m	2.27E-05
1st chamber	0.100%	0.000040	0.0 m	0.00000 m	2.29E-05
Air between chambers	0.329%	0.000076	0.0 m	0.00000 m	4.40E-05
chamber Intrinsic				0.00008 m	1.10E-04
			Quadratic sum	6.09E-04	1.29E-04

Detector Envelopes:

The evolution of the trajectory envelopes are reported in the tables below. In the tables DET x(y) refers to the full vertical (horizontal) extent of the trajectory envelope at the location noted.

	"Normal" 12°	Detector Envelope
		L	DET x	DET y
no cuts		-1.0 m	1.05 m	1.12 m
	~1 m from magnet exit	0.0 m	1.40 m	1.05 m
		1.0 m	1.80 m	0.98 m
		2.0 m	2.20 m	0.93 m
		3.0 m	2.60 m	0.88 m
	calorimeter	4.0 m	3.00 m	0.84 m
		5.0 m	3.40 m	0.80 m

Table 2: Detector Envelope for MAD in the “normal” (quads on) 12° setup.

		No Quads 12°	Detector Envelope
			L	DET x	DET y
no cuts			-1.0 m	1.09 m	1.18 m
		~1 m from magnet exit	0.0 m	1.40 m	1.24 m
			1.0 m	1.85 m	1.32 m
			2.0 m	2.30 m	1.39 m
			3.0 m	2.76 m	1.46 m
		calorimeter	4.0 m	3.21 m	1.54 m
ΔΩ~	4.4 m-sr		5.0 m	3.66 m	1.61 m

θ₀ cut	-0.050		-1.0 m	1.09 m	0.82 m
θ₀ cut	0.035	~1 m from magnet exit	0.0 m	1.37 m	0.85 m
φ₀ cut	-0.022		1.0 m	1.77 m	0.89 m
φ₀ cut	0.022		2.0 m	2.17 m	0.93 m
ΔΩ~	2.4 m-sr		3.0 m	2.57 m	0.96 m
to fit calorimeter		calorimeter	4.0 m	2.97 m	1.00 m
to fit calorimeter			5.0 m	3.37 m	1.03 m

Table 3: Detector Envelope for MAD in the “quadrupoles-off” 12° setup. Both the full envelope and the envelope when it is forced to fit into the 1 x 3 m² Calorimeter are given.

	"Normal" 20°	Detector Envelope
		L