On The Geometry of the EMEC Readout Channels

Local EMEC coordinate system Readout channel geometry parameters Volumes and geometrical centers

Michel Lefebvre University of Victoria Physics and Astronomy

15 August 2003

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 2

The ideal (pointing) ATLAS coordinate system

The , , and quantities for a point in the ideal (pointing) ATLAS coordinate system are defined in the usual way. Using a cylindrical coordinate system, we obtain the following relations:

cossincot

xyz

z

y

x

2 2 2

2 2 2cos

tan

x yz

x y zyx

12ln tan

2arctansinh cotcosh cosectanh cos

e

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 3

EMEC readout channel geometry

For the purpose of volume and geometrical center calculation, the accordion nature of the readout channels is not considered

Three types of readout channels are considered Presampler channels (EMEC layer 0) All other channels, except the

truncated channels in layer 2 The truncated channels in layer 2

TDR fig 7.201 2 3

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 4

EMEC readout channel geometry parameters

For the beam test analysis, the EMEC readout channel geometry parameters are kept in a file, available from http://particle.phys.uvic.ca/~web-atlas/atlas/hec-emec/geometry/

It contains EMEC readout channel “median” coordinates in the ideal (pointing) ATLAS coordinate system:

These quantities do not in general denote the geometrical center of a readout channel. Rather, we have

(in radians) (in cm), , , z z

1 12 2

1 12 2

1 12 2

,

,

,

z z z z z

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 5

Presampler readout channels The volume of a presampler readout channel is considered to be

bounded by two cylinders and by planes at fixed z The z of each channel is set to 1cm in the geometry file From elementary geometry we obtain

12

2 212 12V z

c c

c c

c

cossin

xyz z

where

3 32 1c 2 13 2

1 21 12 2

and

sin

sinh sinh

V z

z z

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 6

Non-truncated EMEC readout channels The volume of a EMEC readout channel (layers 1 to 3, non-

truncated) is considered to be bounded by two cones and by planes at fixed z From elementary geometry we obtain

2 2 2 21a 2 a2 b2 b2 a1 a1 b1 b16V z

c c

c c

c

cossin

xyz

where

a1a2

b2b1

2 2 2 21 1c a 2 b2 a 2 b2 a1 b1 a1 b16 2

2 2 2 21c a 2 b2 a1 b14

sinV z

Vz z z

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 7

Non-truncated EMEC readout channels (continued)

and where

a1a2

b2b1

12

a1 12

12

a2 12

sinh

sinh

z z

z z

12

b1 12

12

b2 12

sinh

sinh

z z

z z

plane z a plane z b

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 8

Truncated EMEC readout channels

These readout channels are in layer 2, and only to the three lowest pseudorapidity bins in 1.400 < || < 1.475

Their volume is bounded by two cones, by a z plane (at z=a) and by a cylinder (m)

z

12

12

a bmz0

m

1m 2

cm

203.37

sinha m b

m

1

2

z m a

z b m

11 2

12 2

z a m

z b m

Let

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 9

Truncated EMEC readout channels (continued)where

a1a2

1 12 2

a1 a2 a11 12 2

sinh sinh

z z z z

plane z a From elementary geometry we obtain

c c

c c

c

cossin

xyz

2 2 21 11 a 2 a2 m m 2 m6 2

2 21a1 a1 m m6

V z z

z

where

2 2 31 1 2 1c 1 a 2 m a 2 m m 26 2 3 2

2 21 1a1 m a1 m6 2

2 2 2 2 21 1 1c 1 1 a 2 m 2 2 m a1 m4 2 4

sin sin

sin

V z z

z

Vz z z z z z z

M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 10

Readout channel volume The following readout channel volumes are obtained

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C-E

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C b

ea

m t

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co

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ura

tion

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M. Lefebvre, 15 August 2003 On the geometry of the EMEC readout channels 11

Readout channel geometrical center The median center can be close to

a cm away from the geometrical center, the difference is (almost completely) in z

c

c

c

0.0019

0.8 cmz z

No difference in z for layer 0 (presampler) since all channels have a cylindrical shape

20

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C-E

ME

C b

ea

m t

est

co

nfig

ura

tion

.