rob 1 intro hill cs tunes - the cockcroft institute · nominal lhc parameters beam injection energy...
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≈ ≈ 40 AE
Nominal LHC parametersBeam injection energy (TeV) 0.45Beam energy (TeV) 7.0Number of particles per bunch 1.15 x 1011
Number of bunches per beam 2808Max stored beam energy (MJ) 362Norm transverse emittance ( m rad) 3.75Colliding beam size ( m) 16Bunch length at 7 TeV (cm) 7.55
Ktr =3
2nRT
d
dtm�x = F (�x, �x, t)
F = −kx
F = −kx
md
dtx = −kx
d2x
dt2= − k
mx
x(t) = x0 sin(ωt+ φ0) ω2 =k
m
md
dtx = −mω2x
x(t) = c1 sin(ωt) + c2 cos(ωt)
x(t) = c1 cosh(ωt) + c2 sinh(ωt)
d2x
dt2= +
k
mx
F = +kx
ω2 =k
m
�F = q( �E + �x× �B)
dxi
dt=
∂H
∂pi
dpidt
= −∂H
∂xi
H =p2x2m
+1
2mω2x2
dx
dt=
pxm
dpxdt
= −mω2x
H = T + V
dH
dt=
∂H
∂x
dx
dt+
∂H
∂px
dpxdt
+∂H
∂t
dH
dt=
∂H
∂x
∂H
dpx− ∂H
∂px
∂H
∂x+
∂H
∂t
dH
dt=
∂H
∂t= 0
x(s)′′ +(k(s) +
1
ρ(s)2
)x(s) = 0
x′′ +(k +
1
ρ2
)x = 0
z′′ − kz = 0
d2u
ds2+Ku(x) = 0
�F = q( �E + �x× �B)
FB = q × 3× 108[m/s]× 1[V s/m2]
B = 1T
FB = q × 300[MV/m]
circular coordinate system
ρ
s
θ
z
FB = qvB
F =γmv2
ρ
qB =γmv
ρBρ =
p
q
circular coordinate system
ρ
s
θ
zBρ =p
q
Bρ =p
q→ 1
ρ=
qB
p
B = 8.3T
p = 7000 GeV/c
1
ρ= q
8.3[Vs/m2]
7000× 109[eV/c]=
8.3 s× 3× 108[m/s]
7000× 109[m2]
1
ρ= 0.33
8.3
7000[/m] = 3.9× 10−4[/m]
E0 >> mc2
Bρ[Tm] =E0[eV]
c[m/s]
k0 =q
p0B
Bρ[Tm] = 3.3 p0[GeV/c]
Bz(x) = Bz0 +dBz
dxx+
1
2
d2Bz
dx2x2 +
1
3!
d3Bz
dx3x3 + ...
e
pBz(x) =
e
pBz0 +
e
p
dBz
dxx+
e
p
1
2
d2Bz
dx2x2 +
e
p
1
3!
d3Bz
dx3x3 + ...
e
pBz(x) =
1
ρ+ kx+
1
2mx2 +
1
3!ox3 + ...
circular coordinate system
ρ
s
θ
z
r = ρ+ x
�v = �R
�v = r�x+ r�x+ z�z
�v = r�x+ rθ�s+ z�z
vx = r vy = z vs = rθ
�x = θ�s �s = −θ�x
y s
r = ρ+ x
rθ = vs
d�p
dt= e�v × �B
d�p
dt=
d
dtmγ �R = mγ �R
r − rθ2 = −eBy
mγvs z =
eBx
mγvs
d�p
dt= mγ(r�x+ 2rθ�s+ rθ�s− rθ2�x+ z�z)
ds
dt= vs
ds
dt= vs
ρ
ρ+ x
r =d2r
dt2=
(vsρ
ρ+ x
)2d2x
ds2
r =dr
dt=
vsρ
ρ+ x
dr
ds=
vsρ
ρ+ x
dx
ds
z =
(vsρ
ρ+ x
)2d2z
ds2
r = ρ+ x
d2z
ds2=
eBx
p
d2x
ds2+
x
ρ2= −e(Bz −Bz0)
p
(ρ
ρ+ x
)2d2x
ds2− 1
ρ+ x= − eBy
mγvs
rθ = vs ΔS = ρΔθ
p = mγv ∼ mγvs
(ρ
x+ ρ
)2
∼ 1− 2x
ρ+ ...
1
ρ+ x∼ 1
ρ− x
ρ2+ ...
d2z
ds2− g
Bρz = 0
d2x
ds2+
(g
Bρ+
1
ρ2
)x = 0
�B = Bz0�z + g(z�x+ x�z)
d2z
ds2− g
Bρz = 0
d2x
ds2+
(g
Bρ+
1
ρ2
)x = 0
d2u
ds2+Ku(x) = 0
ρ = const
k = constx(s)′′ +(k(s) +
1
ρ(s)2
)x(s) = 0
d2u
ds2+Ku(x) = 0
B × leff =
∫ lmag
0
Bds
x′′ +K · x = 0
x(s) = c1 cos(√Ks) + c2 sin(
√Ks)
x′(s) = −c1ω sin(√Ks) + c2ω cos(
√Ks)
x′′(s) = −c1ω2 cos(
√Ks)− c2ω
2 sin(√Ks) = −ω2x(s)
x′′ +(k +
1
ρ2
)x = 0
z′′ − kz = 0
K = k K = k +1
ρ2
K > 0√K = ω
x(0) = x0 → c1 = x0
x′(0) = x′0 → c2 =
x′0√K
K > 0 x(s) = x0 cos(√Ks) + x′
0
1√K
sin(√Ks)
x′(s) = −x0
√K sin(
√Ks) + x′
0 cos(√Ks)
√K = ω
(xx′
)1
= Mquad ·(
xx′
)0
Mfoc quad =
(cos(
√Ks) 1√
Ksin(
√Ks)
−√K sin(
√Ks) cos(
√Ks)
)
s = s0s = s1
Mdefoc quad =
(cosh(
√|K|s) 1√|K| sinh(
√|K|s)+√|K| sinh(√|K|s) cosh(
√|K|s)
)
x′′ −K · x = 0
f(x) = cosh(x) f ′(x) = sinh(x)
x(s) = c1 cosh(√
|K|s) + c2 sinh(√
|K|s)s = s1s = 0
x′′ +(k +
1
ρ2
)x = 0
z′′ − kz = 0
e
p
dBz
dx=
g
Bρ= k
x(s) = x0 + x′0 · L
x′(s) = x′0
Mdrift =
(1 L0 1
)
Mfoc quad =
(cos(
√Ks) 1√
Ksin(
√Ks)
−√K sin(
√Ks) cos(
√Ks)
)
Mthin =
(1 0− 1
f 1
) 1
f= Klq
Mthin =
(1 0− 1
f 1
)
Mfoc quad =
(cos(
√Ks) 1√
Ksin(
√Ks)
−√K sin(
√Ks) cos(
√Ks)
)
Mdefoc quad =
(cosh(
√|K|s) 1√|K| sinh(
√|K|s)+√|K| sinh(√|K|s) cosh(
√|K|s)
)
Mdrift =
(1 L0 1
)
Mcell = MQF ·Mbend ·MQD ·Mdrift ·MQF(xx′
)s=1
= M(s = 1, s = 0) ·(
xx′
)s=0
Q
Mthin =
(1 0− 1
f 1
)
Mfoc quad =
(cos(
√Ks) 1√
Ksin(
√Ks)
−√K sin(
√Ks) cos(
√Ks)
)
Mdefoc quad =
(cosh(
√|K|s) 1√|K| sinh(
√|K|s)+√|K| sinh(√|K|s) cosh(
√|K|s)
)
Mdrift =
(1 L0 1
)
Mfoc quad =
(cos(
√Ks) 1√
Ksin(
√Ks)
−√K sin(
√Ks) cos(
√Ks)
)
K =1
ρ2
θ = l/ρ
1
f=
1
f1+
1
f2− d
f1f2
X(s1) = M(s1|s0)X(s0)
M(s2|s0) = M(s2|s1)M(s1|s0)
M(s+ C|s)
M(s+ C|s)N
20 25[ km ]
IP6 IP7 IP8 IP
Horizontal trajectory
Injected beam + beam after one turn on screen
k(s+ L) = k(s)
x(s)′′ +(k(s) +
1
ρ(s)2
)x(s) = 0
x(s) =√ε β(s) cos(ψ(s) + ψ0)
β(s+ L) = β(s)
1
2(ββ′′ − 1
2β′2)− β2ψ2 + β2k = 0
β′ψ′ + βψ′′ = 0
β′ψ′ + βψ′′ = (βψ′)′
βψ′ = 1
ψ(s) =
∫ s
0
ds
β(s)
1
2ββ′′ − 1
4β′2 + β2k = 1
α(s) = −1
2
dβ(s)
dsγ(s) =
1 + α2(s)
β(s)
x(s) =√
ε β(s) cos(ψ(s) + ψ0)
x′(s) = −√ε√β
[α(s) cos(ψ(s) + ψ0) + sin(ψ(s) + ψ0)
]
βx′ + αx = −√
εβ sin(ψ + ψ0)
x2 + (βx′ + αx)2 = εβ
γx2 + 2αxx′ + βx′2 = ε
0
γx2 + 2αxx′ + βx′2 = ε
γx2 + 2αxx′ + βx′2 = ε
A = πε
= 55 cm = 16 m; D = 0m
R. Tomàs
β(0) = β0 α(0) = α0 ψ(0) = 0
c1 =x0√β0
c2 =√
β0x′0 +
α0√β0
x0
x(s) =
√β(s)
β0[cosψ(s) + α0 sinψ(s)]]x0 +
√β0β(s)x
′0 sinψ(s)
x(s) =
√β(s)
β0[cosψ(s) + α0 sinψ(s)]]x0 +
√β0β(s)x
′0 sinψ(s)
(x(s1)x′(s1)
)= M(s1|s0)
(x(s0)x′(s0)
)
ψ = ψ(s1)− ψ(s0)
M(s+ L|s) =(
cosΨ + α sinΨ β sinΨ−γ sinΨ cosΨ− α sinΨ
)
ψ1 − ψ0 = Ψ
β1 = β0 = β α1 = α0 = α γ1 = γ0 = γ
γ(s) =1 + α2(s)
β(s)
Ψ = arccos
(m11 +m22
2
)
β =m12
sinΨα =
m11 −m22
2 sinΨγ = − m21
sinΨ
|TrM | ≤ 2
M =
(m11 m12
m21 m22
)
γ(s) =1 + α2(s)
β(s)
M(s′ + C|s′) = M(s′|s) ·M(s+ C|s) ·M−1(s′|s)
M =
(m11 m12
m21 m22
)
⎛⎝ α1
β1
γ1
⎞⎠ =
⎛⎝ m11m22 +m12m21 −m11m21 −m12m22
−2m11m12 m211 m2
12
−2m21m22 m221 m2
22
⎞⎠
⎛⎝ α0
β0
γ0
⎞⎠
M(s′ + C|s′) = M(s′|s) ·M(s+ C|s) ·M−1(s′|s)
M(s+ L|s) =(
cosΨ + α sinΨ β sinΨ−γ sinΨ cosΨ− α sinΨ
)
Mdrift =
(1 L0 1
)
m11 = 1 m12 = L m21 = 0 m22 = 1
α1 = α0 − γ0L
β1 = β0 − 2α0L+ γ0L2
γ1 = γ0
x(L) = x0 + Lx′0
x′(L) = x′0
Ψ =
∫ s+L
s
ds
β(s)
ν =Ψ
2π=
∫ s+C
s
ds
β(s)