Prof. Dr. Johanna Stachel
Department of Physics und Astronomy
University of Heidelberg
June 17, 2015
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 1 / 333

6 Momentum Measurements
Forward Spectrometer
Solenoidal and Toroidal Fields - mostly at Colliders
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Momentum Measurements
Deflection of track of charged particle in magnetic spectrometer
Lorentz force → circular orbit of curvature radius ρ in homogeneous magnetic field mv
2
ρ
= q ~ × p
2
ρ = qp
⊥
B
~
= qv
⊥
· |
~
| and for ~ ⊥ B p
ρ = qB v p
⊥
⊥
: component of
: analogue
~ ⊥ to units: for ρ in m p in GeV/c
B in T q in units of e p
ρ =
0 .
3 qB or p = 0 .
3 q ρ B
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 3 / 333

Momentum Measurements Forward Spectrometer
Mainly in fixed target experiments (but also LHCb) or ALICE forward muon spectrometer
L
_
_ beam x z
Target tracking chambers dipole magnet
B = (0, B y
+
, 0)
+ magnetic field gives (additional) p
⊥ typically p p
⊥
, ∆ p
⊥
-kick ∆ p
⊥
→ Lorentz force always approximately in x -direction and
∆ p x
= 0 .
3 L q B or for magnetic field not constant over entire path
∆ p x
= 0 .
3 q
Z
L
B d L
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Momentum Measurements Forward Spectrometer
Dipole magnet
Muon chambers
J. Stachel (Physics University Heidelberg) Detectorphysics
Muon absorber and filter
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Momentum Measurements Forward Spectrometer
Example: proton of p = 10 GeV/c ' p z
R
BdL = 6 Tm
∆ p x
= 1.8 GeV/c
∆ θ x
= 10
◦ about the limit for this approximation
ρ x
θ/
2 z
θ h for ρ L or
θ ≈
L
ρ
=
LqB y p
∆ p x
= p sin θ ≈ p θ = LqB y
≈ q
Z
L
0
B y d L
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 6 / 333

Momentum Measurements Forward Spectrometer
p = q ρ B y
= q
L
B y
θ dp d θ
1
= qLB y
θ 2
= p
θ
⇒ dp
= p
σ p
= p d
σ
θ
θ
θ
θ accuracy of angular measurement ≡ accuracy of momentum measurement minimum: two measured points before and two after deflection in practice always 3 or more measurements, since detectors need to be aligned relative to each other (best done with straight tracks) in case all measured points have identical resolution σ x
: n / 2 points before n / 2 points after deflection
⇒ σ
θ
= r
8 n
σ x h lever arm h (see Fig. previous page)
σ p p
= p
8 / n σ x h p qLB y
= p
8 / n σ x h p
∆ p x contribution of space point resolution to momentum resolution typical form
σ p
= const · p with const = 10
− 2 . . .
10
− 4 i.e.
1% - 0.01% p
Example: 6 measurements each with σ x
= 200 µ m, h = 5 m, deflection 1
◦
⇒
σ
θ x
=
∆ p x p
= 0 .
017 deflection) for p = 10 GeV / c
= 3 · 10
− 3
= 3 · 10
− 4 p p p
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 7 / 333

Momentum Measurements Forward Spectrometer
multiple Coulomb scattering along particle trajectory of length L contributes to p
⊥
-broadening perpendicular to direction of propagation
∆ p ms
⊥
= p sin θ rms
' p θ rms
= q · 19 .
2 MeV/c s
L
β X
0 where X
0 is the radiation length. In the direction of deflection ( x ) this means:
∆ p x ms
= q · 19 .
2 MeV/c
√
β 2 s
L
X
0
= q · 13 .
6 MeV/c s
β
L
X
0 for sufficiently large momenta independent of p contribution to momentum resolution:
σ p p ms
=
∆ p ms x
∆ p x
=
13 .
6 MeV/c p
L / X
0 e
R
B y d L where ∆ p x is the deflection due to magnetic field (see above).
J. Stachel (Physics University Heidelberg) Detectorphysics
Δ p ms x
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Momentum Measurements Forward Spectrometer
Example: as above e
Z
BdL = ∆ p x
= 0 .
17 GeV/c
L = 15 m material: air, X
0
= 304 m vs .
σ p p
σ p p ms defl
= 1
= 0 .
.
8%
03% p ⇒ multiple scattering dominates at small momenta
σ p p
2
=
σ p p
2 ms
+
σ p p
2 defl momentum resolution in magnetic spectrometer
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 9 / 333

Example:
Momentum Measurements Forward Spectrometer
σ p p
2
= (1 .
8%)
2
+ (0 .
06% · p )
2
σ p p
[%]
3.
2.
1.
J. Stachel (Physics University Heidelberg)
10
Detectorphysics both together
σ p p
2
σ p p defl
20 p [GeV/c]
σ p p
2 ms
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Momentum Measurements Forward Spectrometer
Multiple scattering particularly relevant if magnetized iron is used, as frequently done for measurements of muon momentum advantage: high B -field, stops π, K before they decay into µ disadvantage: worsens momentum resolution by multiple scattering
X
0
Fe
= 1 .
76 cm ,
⇒
B = 1
σ p p
.
8 T ms
,
= 11%
L = 3 m , ∆ p x
= 1 .
6 GeV/c depending on desired momentum range, accuracy of deflection measurement can be chosen accordingly
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
B
Normally 4 π coverage desired, leading to special spectrometer configuration dipole disfavored
deflects beam which must be compensated
not nice symmetry for 4 π experiment p
Solenoid long cylindrical coil
I p beam axis
B
B along beam direction, no deflection measure momentum component p
⊥ perpendicular to beam
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 12 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders y beam and B -field along z -axis particle produced with momentum ~ completely characterized by p x
, p y
, p z
, where p x and p y can be written as:
| p
⊥
| , ϕ : p x
= | p
⊥
| cos ϕ p y
= | p
⊥
| sin ϕ p
φ
x
need to measure at least 3 points of track circular in xy -plane
⇒ ρ (radius of curvature) or p
⊥ and ϕ r measurement of θ : p k
= p = p
⊥ tan θ p
⊥ sin θ p
θ p p z complete measurement of particle momentum
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 13 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
x
θ/
2
S
L
ρ
Sagitta S = ρ − ρ cos
θ
2
θ
= 2 ρ sin
2
4
ρθ
2 for small θ S '
8
= ρ 1 − cos
θ
2 with θ = qBL p
⊥ and sin θ/ 2 ' θ/ 2 =
L / 2
ρ
S = qL
2
B
8 p
⊥
B in T, L in m, p
⊥ in GeV/c,
S (m) =
0 .
3 qL
2
B
8 p
⊥ q in e
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 14 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
Measurement of at least 3 points with coordinates x
1
, x
2
, x
3
σ
S = x
2
− x
1
+ x
3
2
S
= r
3
2
σ x
⇒
σ p p
=
σ
S
S
= p
3 / 2 σ x
8 p
0 .
3 qBL 2
Measurement of N equally spaced points:
σ p p
=
σ x
0 .
3 qBL 2 s
720
( N + 4) p
1
NIM 24 (1963) 381 example
B = 0 .
5 T
L = 2 m
σ x
= 400 µ m
N = 150



σ p p


' 1 .
4 · 10
− 3 p similar to ALICE TPC, usage of former L3 Magnet
2
J. Stachel (Physics University Heidelberg) Detectorphysics
3
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
σ
S
= 90 µ m
J. Stachel (Physics University Heidelberg)
σ p
⊥ p
⊥
= 2 .
5% at 45 GeV (typical Z
0 decay product)
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
Largest magnet, about 7000 t iron (like Eiffel tower) current in magnet 30 kA ← average lightning energy stored in magnetic field equivalent to explosion of 45 kg TNT bomb
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 17 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 18 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
B
I on axis vanishing B -field no deflection of beam fill with iron-core e.g. for muon measurement in end caps
Example: H1 forward muon spectrometer
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 19 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
Central solenoid plus forward muon toroid to measure high energy muons between 3
◦ and 17
◦ drift chamber planes before and after toroid
J. Stachel (Physics University Heidelberg) Detectorphysics
Toroidal magnet: 12 segments with
15 turns each (Cu), 150 A
→ B = 1 .
6 T filled with Fe core
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12 m
J. Stachel (Physics University Heidelberg)
Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
- ro = 2.9 m
- r i
= 0.65 m deflection in polar angle → momentum
σ p
= 24 − 36% p for p = 5 − 200 GeV/c dominated by multiple scattering
σ p
= 0 .
24 ⊕ 1 .
3 · 10
− 3 p p
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 22 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
‘air core toroid’ central barrel
L = 26 m, D i
= 9.4 m, D
0
= 19.5 m
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 23 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
8 flat coils, super-conducting, 70 km super-conducting cable
R
20 kA, BdL = 3 − 9 Tm energy stored in magnetic field 1490 MJ
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
F with ’current on’ forces on coils radially inward
B -field monitored by 5000 Hall probes attached to muon chambers
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η = 2.0
η = 2.4
η = 3.0
η = 1.4
η = 1.05
Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
Principle of momentum measurement of muon tracks with monitored drift-tube array:
3 layers, each consisting of
2 multilayers total 1200 muon chambers of
2 × 3 .
5 m
2 total 300000 channels
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
drift tubes Al-Mn
∅
3 cm
400 µ m wall thickness
1 .
4 − 6 .
3 m long 50 µ m wire centered in tube to 20 µ m operation at 3 bar drift time ≤ 600 ns gas gain < 50000
(only ’streamers’ to avoid ↔ ’aging’) gas: Ar/C
2
H
6
/CO
2
/N
2
86 : 5 : 4 : 5
Cross plate
Multilayer
In-plane alignment
Longitudinal beam
J. Stachel (Physics University Heidelberg) Detectorphysics June 17, 2015 28 / 333

Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders position resolution ATLAS monitored drift tube array difficulty: gravitational sag of long tubes and wires
J. Stachel (Physics University Heidelberg) Detectorphysics single tube position resolution 80 µ m
→ σ p t
/ p t
= 10% at 1000 GeV
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Momentum Measurements Solenoidal and Toroidal Fields - mostly at Colliders
Rasnick System
13000 CCD cameras
→ align each muon chamber to 0.05 mm precision
J. Stachel (Physics University Heidelberg) Detectorphysics
Alignment system for 3 layers of MDT’s
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