MiniBooNE: Status and Prospects Eric Prebys, FNAL/BooNE Collaboration.
Eric Prebys, FNAL
Transcript of Eric Prebys, FNAL
Eric Prebys, FNAL
Ø Math Refresher (Expectations) Ø Maxwell’s Equations Ø Special Relativity Ø Multipole Expansion of Magnetic Fields
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 2
Ø Matrix Operations
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 3
a bc d
!
"#
$
%&
V1V2
!
"
##
$
%
&&=
aV1 +bV2cV1 +dV2
!
"
##
$
%
&&
a bc d
!
"#
$
%&
−1
=1
ad −bcd −b−c a
!
"#
$
%&
a bc d
≡ det a bc d
"
#$
%
&'= ad −bc( )
a b cd e fg h i
= a e fh i
−bd fg i
+ c d eg h
Ø Vector Operations u Dot product
u Cross product
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 4
!A ⋅!B = (AxBx + AyBy + AzBz )
!A×!B =
i j kAx Ay AzBx By Bz
= (AyBz − AzBy )i + (AzBx − AxBz ) j + (AxBy − AyBx )k
Ø Vector differential operations u Grad operator
u Gradient
u Divergence
u Curl
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 5
∇!"≡
∂∂xi + ∂
∂yj + ∂
∂zk
$
%&
'
()
∇!"φ ≡
∂φ∂xi + ∂φ
∂yj + ∂φ
∂zk
%
&'
(
)*
∇!"×"A ≡
i j k∂∂x
∂∂y
∂∂z
Ax Ay Az
=∂Az∂y
−∂Ay
∂z&
'(
)
*+ i +
∂Ax
∂z−∂Az∂x
&
'(
)
*+ j +
∂Ax
∂y−∂Ay
∂x&
'(
)
*+ k
∇!"⋅"A ≡ ∂Ax
∂x+∂Ay
∂y+∂Az∂z
%
&'
(
)*
Ø You should be very comfortable with the complex plane
Ø Also remember the Taylor expansions of trig functions
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 6
eiθ = cosθ + isinθ
cosθ = eiθ + e−iθ
2
sinθ = eiθ − e−iθ
2i
eθ ≈ 1+θ +θ2
2!+θ 3
3!+ ...
sinθ ≈θ −θ3
3!+θ 5
5!− ...
cosθ ≈ 1−θ2
2!+θ 4
4!− ...
Ø Memorize these because we’ll use them a lot!
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 7
( )
( )
( )
( )
( )
( ))2cos(121sin
)2cos(121cos
)cos()cos(21sinsin
)cos()cos(21coscos
)sin()sin(21sincos
)sin()sin(21cossin
1cos22coscossin22sin
sinsincoscos)cos(sinsincoscos)cos(sincoscossin)sin(sincoscossin)sin(
2
2
2
AA
AA
BABABA
BABABA
BABABA
BABABA
AAAAA
BABABABABABABABABABABABA
−=
+=
+−−=
−++=
−−+=
−++=
−=
=
+=−
−=+
−=−
+=+
Ø In 1861, James Maxwell began his attempt to find a self-consistent set of equations consistent with all of the E&M experiments which had been done up until that point. u Because vector calculus hadn’t been invented yet, his final paper is
55 pages long and completely incomprehensible.
Ø In in modern notation, it reduces to the following four equations:
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 8
!∇•!E =
ρε0
⇒!E •d!A
S"∫ =Qenc
ε0
Gauss' Law
!∇•!B = 0 ⇒
!B•d!A
S"∫ = 0 No Name Law
!∇×!E = −
∂!B∂t
⇒!E •d!l
C"∫ = −∂∂t
!B•d!A
S"∫ Faraday's Law
!∇×!B = µ0
!J +µ0ε0
∂!E∂t
⇒!B•d!l
C"∫ = µ0Ienclosed +µ0ε0∂∂t
!E •d!A
S"∫ Ampere's Law
Ø The electric field passing through a surface depends only on the charge contained within the surface
Ø Example: deriving Coulomb’s Law
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 9
!E •d!A
S"∫ =Qenc
ε0
!E •d!A
S"∫ = E •A
= 4πr2E
=qε0
→ E = q4πr2ε0
!B•d!A
S"∫ = 0→No magnetic monopoles
Ø The integrated electric field around any closed loop is proportional to the rate of change of the magnetic flux passing through the loop
Ø Example: magnetic induction
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 10
!E •d!l
C"∫ = −∂∂t
!B•d!A
S"∫
V =!E •d!l
C"∫= −
∂∂t
!B•d!A
S"∫
= −B dAdt
= −Bwv
w
Ø The integrated magnetic field around any closed loop is proportional to the total current passing through the loop.
Ø Example: Magnetic field of a wire
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 11
!B•d!l
C"∫ = µ0Ienclosed +µ0ε0∂∂t
!E •d!A
S"∫Set to 0 for a minute
!B•d!l
C"∫ = 2πrB
= µ0Ienclosed = µ0I
→ B = µ0I2πr
Ø Maxwell’s first version of Ampere’s Law did not have the second term
Ø However, you should be able to draw the surface anywhere, and you get in trouble if you draw it through a break in the current
Ø Maxwell added the second term just so he would get the same answer in both cases!
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 12
!B•d!l
C"∫ = µ0Ienclosed
Current flowing here
Field changing here
However, anywhere there’s a break in the current, you’ll get a changing electric field.
!B•d!l
C"∫ = µ0Ienclosed +µ0ε0∂∂t
!E •d!A
S"∫
Ø The “displacement current” was added for purely mathematical reasons u It would not be proven experimentally for many years
Ø However, the implications were profound Ø Previously, it was believed you could not have electric or magnetic
fields without electric charges, but now, even in a complete vacuum, you can have u (changing electric field)è(changing magnetic field)è�
(changing electric field)è“Electromagnetic Wave”! Ø Moreover, Maxwell could calculate the velocity,
and he found it was the speed of light! Ø He wrote (with trembling hands, maybe?)
"we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena"
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 13
Ø In one fell swoop, Maxwell not only unified electricity and magnetism, but his results would eventually show that light, heat, radio waves, x-rays, gamma rays, etc., are all really the same thing – differing only in wavelength!
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 14
The entire visible spectrum.
Ø As often happens science, one answer raised a lot more questions.
Ø All (other) known waves require a “medium” (air, water, earth, “the wave”) to travel through.
Ø Light at least appears to travel through a vacuum. Ø In science, always try the simplest answer first:
u Maybe vacuum isn’t really empty?
Ø Scientists hypothesized the existence of “luminiferous aether”, and started to look for it…
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 15
Ø If aether exists, then it must fill space and the earth must be passing through it.
Ø Light traveling along the direction of the Earth’s motion should have a slightly different wavelength than light traveling transverse to it.
Ø In 1887, Albert Michelson and Edward Morley performed a sensitive experiment to measure this difference.
Ø Their result: u No difference è no aether!
Ø Biggest mystery in science for almost 20 years.
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 16
Ø In 1905, Albert Einstein postulated that perhaps the equations meant exactly what they appeared to mean: u The speed of light was the same in any frame in which is was
measured.
Ø He showed that this could “work”, but only if you gave up the notion of fixed time. u è “Special Theory of Relativity”
Ø Profound implications…
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 17
Ø Einstein said, “The speed of light must be the same in any reference frame”. For example, the time it takes light to bounce off a mirror in a spaceship must be the same whether it’s measured by someone in the spaceship, or someone outside of the spaceship.
Ø This seems weird, but it applies to everything we do at the lab u Example: the faster pions and muons move, the longer they live.
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 18
• These two people have to measure the same speed for light, even though light is traveling a different distance for the two of them.
• The only solution? More time passes for the stationary observer than the guy in the spaceship!
• “Twin Paradox”
Ø Generally, relativity treats time more or less like one more spatial dimension. Both time and space transform between two frames
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Ø Classically:
Ø Relativistically:
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momentum: !p = m!v
kinetic energy: K =12mv2
momentum: !p = m!v1− (v / c)2
total energy: E2 = (mc2 )2 + (pc)2
kinetic energy: K = E-mc2
Emc2
pc
0"
1"
2"
3"
4"
5"
6"
7"
8"
0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1"
Rela%v
is%c*"gamma"*fa
ctor*
(velocity)/(speed*of*light)*
Rest Energy
Kinetic
Energy
For v<<c (speed of light),
Kinetic energy ~ ½mv2
γ = 1
1− vc
⎛⎝⎜
⎞⎠⎟2
c = (speed of light) = 300,000 km/s!
Ø Basics
Ø A word about units u For the most part, we will use SI units, except
u Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J] u Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2] u Momentum: eV/c [proton @ β=.9 = 1.94 GeV/c]
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 21
β ≡vc
γ ≡1
1−β 2
momentum p = γmvtotal energy E = γmc2
kinetic energy K = E −mc2
E = mc2( )2+ pc( )2
β = pcE
dγ = βγ 3dβdββ
= 1γ 2
dpp
dpp
= 1β 2
dEE
Some Handy Relationships (homework)
Ø We’ll use the conventions
Ø Note that for a system of particles
Ø We’ll worry about field transformations later, as needed
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 22
( )
( ) ( )
( )222222
2
222222
axis) z along(velocity
0000
00100001
,,,
,,,
mcpppcE
czyxct
cEppp
ctzyx
zyx
zyx
≡−−−⎟⎠
⎞⎜⎝
⎛=
≡−−−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−==ʹ′
⎟⎠
⎞⎜⎝
⎛≡
≡
P
X
AΛAA
P
X
τ
γβ
βγ
( ) scMeffi ≡=∑222
P
Ø The equations we’ve talked about so far are correct if you account for all electric charges in the system; however, in real life situation, much, or even most, of the charge is a system is contained in matter, and it’s behavior can generally be parameterized in a more convenient way. In terms of just the free electric charge, Gauss’ Law and Ampere’s Law become: where
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 23
!∇•!D = ρ f ⇒
!D•d
!A
S"∫ =Qf ,enc ;!D ≡ε
!E
!∇×!H =
!J f +
∂!D∂t
⇒!H •d
!l
C"∫ = I f ,enclosed +0∂∂t
!D•d
!A
S"∫ ;!H ≡
!Bµ
Local effects of media
ε = "electric permitivity"µ = "magnetic permiability"
Ø The “electric permittivity” comes from the tendency of charge in matter to form electric dipoles in the presence of an external field, reducing the the true field
Ø The “magnetic permeability” comes from the tendency of magnetic dipoles in some materials to align with the external magnetic field, increasing the true field.
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 24
Ø Cross section of dipole magnet
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 25
g
Integration loop
!H •d
!l
C"∫ =
1µsteel
!B•d!l
path in steel∫ +
Bgapgµ0
≈Bgapgµ0
= Ienclosed
gINB turns
gap0µ≈⇒
µsteel µgap
Ø The relativistially correct form for the motion of charged particles in electric and magnetic fields is given by the Lorentz equation:
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 26
F = d
!pdt
= q(!E + !v ×
!B)
radius of curvature r = pqB
Ø A charged particle in a uniform magnetic field will follow a circular path of radius
side view
B
ρ
top view
Bρ =
mvqB
(v << c)
f =v
2πρ
=qB
2πm (constant!!)
Ωs = 2π f = qBm
MHz ][2.15 TBfC ×=
“Cyclotron Frequency”
For a proton:
Accelerating “DEES” 27
Ft. Collins, CO, June 13-24, 2016
E. Prebys, Accelerator Fundamentals: Basic EM and Relativity
Ø The relativistically correct form of Newton’s Laws for a particle in an electromagnetic field is:
Ø A particle of unit charge in a uniform magnetic field will move in a circle of radius
!F = d
!pdt
= q!E + !v ×
!B( ); !p = γm!v
ρ = peB
Bρ( ) = pe
Bρ( )c = pce
side view
B
ρ
top view
Bconstant for fixed energy!
T-m2/s=V units of eV in our usual convention
Bρ( )[T-m]= p[eV/c]c[m/s]
≈ p[MeV/c]300
Beam “rigidity” = constant at a given momentum (even when B=0!)
Remember forever!
If all magnetic fields are scaled with the momentum as particles accelerate, the trajectories remain the same è“synchrotron” [E. McMillan, 1945]
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 28
Ø Compare Fermilab LINAC (K=400 MeV) to LHC (K=7000 GeV)
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 29
Parameter Symbol Equa0on Injec0on Extrac0on proton mass m [GeV/c2] 0.938 kine9c energy K [GeV] .4 7000 total energy E [GeV] 1.3382 7000.938 momentum p [GeV/c] 0.95426 7000.938 rel. beta β 0.713 0.999999991
rel. gamma γ 1.426 7461.5 beta-‐gamma βγ 1.017 7461.5
rigidity (Bρ) [T-‐m] 3.18 23353.
K +mc2
E2 − mc2( )2
pc( ) / EE / (mc2 )
p[GeV]/(.2997)pc( ) / (mc2 )
This would be the radius of curvature in a 1 T magnetic field or the field in Tesla
needed to give a 1 m radius of curvature.
Ø If the path length through a transverse magnetic field is short compared to the bend radius of the particle, then we can think of
the particle receiving a transverse “kick” and it will be bent through small angle
Ø In this “thin lens approximation”, a dipole is the equivalent of a prism in classical optics.
lB θΔ
p
)( ρθ
BBl
pp
=≈Δ ⊥
qBlvlqvBqvBtp ==≈⊥ )/(
θΔ
Ft. Collins, CO, June 13-24, 2016 30 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity
Ø Define the “gradient” operator
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 31
∇!"≡∂∂xi + ∂
∂yj + ∂
∂zk
∇!"⋅"A = ∂Ax
∂x+∂Ay
∂y+∂Az∂z
∇!"×"A = ∂Az
∂y−∂Ay
∂z'
()
*
+, i +
∂Az∂x
−∂Ax
∂z'
()
*
+, j +
∂Ay
∂x−∂Ax
∂y'
()
*
+, k
=
i j k∂∂x
∂∂y
∂∂z
Ax Ay Az
Ø Formally, in a current free region, the curl of the magnetic field is:
Ø This means that the magnetic field can be expressed as the gradient of a scalar:
Ø The zero divergence then gives us:
Ø If the field is uniform in z, then δφ/δz=0, so
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 32
!∇×!B = µ0
!J = 0
Laplace Equation !B = −∇
"!φ
∇!"⋅"B = −∇2φ =
∂2φ∂x2
+∂2φ∂y2
+∂2φ∂z2
&
'(
)
*+= 0
∂2ϕ∂x2
+∂2ϕ∂y2
= 0
Ø The general solution is
Ø Solving for B components
Ø Combining and redefining the constants
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 33
Bx = −∂ϕ∂x
= −Re mCm x+ iy( )m−1m=1
∞
∑ = −Im imCm x+ iy( )m−1m=1
∞
∑
By = −∂ϕ∂y
= −Re imCm x+ iy( )m−1m=1
∞
∑
By + iBx = Kn x+ iy( )nn=0
∞
∑ ;Kn = i(n+1)Cn+1
∂2ϕ∂x2
+∂2ϕ∂y2
= 0⇒ϕ (x, y)=Re Cm x+ iy( )mm=0
∞
∑
Note order!
Ø We can express the complex numbers in notation
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 34
By + iBx = Kn x+ iy( )nn=0
∞
∑ = Knrn
n=0
∞
∑ einθ
= Kn eiδn rn
n=0
∞
∑ einθ
Amplitude rotation
r is real Kn is complex
Ø In our general expression the phase angle δm represents a rotation of each component about the z axis. Set all δm =0 for the moment, and we see the following symmetry properties for the first few multipoles
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 35
),()2/,(0)4/,(;)4/,(
sextupole)0,(;0)0,(2),(),(0)2/,(;)2/,(
quadrupole)0,(;0)0,(1dipole;00
,,
22
22
,,
1
1
0
θπθ
ππ
θπθ
ππ
rBrBrBKrrB
KrrBrBnrBrB
rBKrrBKrrBrBn
KBBn
yxyx
yx
yx
yxyx
yx
yx
yx
−=+
==
≡==⇒=
−=+
==
≡==⇒=
≡==⇒=
By + iBx = Kn eiδn rn
n=0
∞
∑ einθ
Ø Back to Cartesian Coordinates. Expand by differentiating both sides n times wrt x
Ø And we can rewrite this as
Ø “Normal” terms always have Bx=0 on x axis. Ø “Skew” terms always have By=0 on x axis. Ø Generally define
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 36
( )
nyx
nx
n
yxny
nn
nnxy
KnxBi
xB
iyxKiBB
!00
0
=⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
∂
∂+
∂
∂⇒
+=+
====
∞
=∑
( )( )
0
00
~
;~!
1
==
==
∞
=
∂
∂≡
∂
∂≡++=+ ∑
yxxn
n
n
yxyn
n
nn
nnnxy
Bx
B
Bx
BiyxBiBn
iBB“normal”
“skew”
etc ,~~,~~,, 2121 BBBBBBBB ≡ʹ′ʹ′≡ʹ′≡ʹ′ʹ′≡ʹ′
Ø Expand first few terms…
Ø Note: in the absence of skew terms, on the x axis
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 37
( )
( ) ...2
~~~
...~2
~
220
220
+ʹ′ʹ′+−ʹ′ʹ′
+ʹ′+ʹ′+=
+ʹ′ʹ′−−ʹ′ʹ′
+ʹ′−ʹ′+=
xyByxByBxBBB
xyByxByBxBBB
x
y
dipole quadrupole sextupole
nny x
nBxBxBxBBB!
...62
320 +
ʹ′ʹ′ʹ′+
ʹ′ʹ′+ʹ′+=
dipole quadrupole sextupole octupole
Ø Dipoles: bend Ø Quadrupoles: focus or defocus
Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 38
� A positive particle coming out of the page off center in the horizontal plane will experience a restoring kick
xB
y
yB
x
)()()(
ρρθ
BlxB
BlxBx ʹ′
−=−≈Δ
lBBf')( ρ
=
∇!"×"B = 0
→∂By
∂x=∂Bx
∂y
Ø Sextupole magnets have a field (on the principle axis) given by
Ø One common application of this is to provide an effective position-dependent gradient.
Ø In a similar way, octupoles have a field given by
Ø So high amplitude particles will see a different average gradiant
2
21)( xBxBy ʹ′ʹ′=
x
yB
x
BxBeff ʹ′ʹ′=ʹ′
Ft. Collins, CO, June 13-24, 2016 39 E. Prebys, Accelerator Fundamentals: Basic EM and
Relativity
3
61)( xBxBy ʹ′ʹ′ʹ′=
x
yB
maxx
Bx
Beff ʹ′ʹ′ʹ′=ʹ′2
2max