Limit on a horizontal emittance in high energy muon colliders due to synchrotron radiation

2 Pages·0.05 MB·English

by  Valery Telnov

Limit on a horizontal emittance in high
energy muon colliders due to synchrotron
radiation.
∗
0
0
0
Valery Telnov,
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n Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
†
a
J

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1

Abstract
1
v
It is shown that at a 100 TeV muon collider the synchrotron radiation in the
2
3 ring will determine the minimum horizontal emittance.
0
1 In all e+e− storage ring the horizontal emittance is determined by quantum nature
0
0 of the synchrotron radiation. Emission of photons in regions with non-zero dispersion
0
leads to growth of the horizontal emittance while the average energy loss leads to the
/
x
cooling. As the result, some equilibrium emittance is reached. This is well known and
e
- corresponding formulae can be found elsewhere [1].
p
At muon colliders, thesynchrotron radiationpower is much smaller thanthat at e+e−
e
h storage rings due to higher particle mass: P (EB)2/m4; however this suppression is
: ∝
v compensated by the much higher energy and magnetic field.
i
X In general, emittance in a storage ring depends on time in the following way
r
a ǫ = ǫ (1 e−t/τ)+ǫ e−t/τ, (1)
x D 0
−
where ǫ is the initial emittance, ǫ is the equilibrium emittance, τ is the damping time.
0 D
One can check (see also the B.King’s table) that for the 2E=100 TeV muon collider
γ τ τ 1 sec, so the emittance will be close to the equilibrium one.
µ µ D
≈ ≈
The equilibrium normalized (ǫ = γǫ) horizontal emittance in a damping ring [1]
n
55h¯ γ3 H
ǫ = h i, (2)
D
32√3mcJ ρ
x
∗Talk at the Workshop Studies on Colliders and Collider Physics at the Highest Energies: Muon
Collidersat10TeVto100TeV,27September-1October,1999Montauk,NewYork,USA,bepublished
by the American Institute of Physics.
†email:[email protected]
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where J 1 is the damping partition number, H β3/ρ2 is the average value of
x
≈ h i ∝
the “H-function” characterizing a storage ring magnetic structure. One can see that for
fixed energy and magnetic field ǫ 1/m.4
nx
∝
For an optimized FODO ring structure of the muon collider the minimum normalized
horizontal emittance due to synchrotron radiation [1]
E3[GeV] m 4
e
ǫ 100 cm rad, (3)
D ∼ N3 mµ!
where N is the number of bending magnets in the ring. Extrapolating the number of
magnets in the current designs to the 100 TeV energy as N √E (the usual energy
∝
dependenceoftheβ-function)onecangetN 1000for2E = 100TeV.Forthisstructure
∼
we obtain the normalized horizontal emittance at t = τ
D
−3
ǫ 0.63ǫ 5 10 cm rad. (4)
nx D
≈ ≈ ×
For comparison, in the B.King’s table ǫ = 8.7 10−4 cm rad for “evolutionary”
nx
extrapolation and ǫ = 2.1 10−5 cm rad for “ultra-×cold beams”, which is smaller than
nx
×
above result by factors of 5 and 250, respectively.
This example shows that the considered effect is important for high energy muon
colliders and this problem needs more accurate consideration. One should consider
the more realistic case of the isochronous ring (which is necessary for muon colliders),
including W-shielding which reduces the quad strength (and correspondingly N).
References
[1] H.Wiedemann, Particle Accelerator Physics, v.1, Springer-Verlag.
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