Basic Maths and Physics for Accelerators
Transcript of Basic Maths and Physics for Accelerators
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
C. Biscari
Basic Maths and Physics for
Accelerators
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 2
Physical constants
Constant Symbol Value
Speed of light in vacuum c 2.99792458 108 m/s
Electric charge unit e 1.60217653 10-19 C
Electron rest energy mec2 0.51099818 MeV
Proton rest energy mpc2 938.27231 MeV
Fine structure constant a 1/137036
Avogadroβs number A 6.021415 1023 1/mol
Classical electron radius rc 2.8179433 10-15 m
Planckβs constant h 4.1356675 10-15 eV s
l of 1 eV photon Δ§c/e 12398.419 Θ¦
Permittivity of vacuum o 8.85418782 10-12 C/(Vm)
Permeability of vacuum o 1.25663706 10-6 Vs/(Am)
SI measurement system
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Vector calculus
Gradient of a scalar function
π π₯, π¦, π§, π‘
π»π =ππ
ππ₯,ππ
ππ¦,ππ
ππ§
Divergence
π» β πΉ =ππΉ1
ππ₯+ππΉ2
ππ¦+ππΉ3
ππ§
For a vector
πΉ = πΉ1, πΉ2, πΉ3
Curl
π» Γ πΉ
=ππΉ3ππ¦
βππΉ2ππ§,ππΉ1ππ§
βππΉ3ππ₯,ππΉ2ππ₯βππΉ1ππ¦
Remember for example:
A vector potential is a vector field whose curl is a given vector field. A scalar potential is a scalar field whose gradient is a given vector field
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 4
Maxwellβs Equations
Relate electric and magnetic fields generated by charge and current distributions Put together in 1863 results known from work of Gauss, Faraday, Ampere, Biot, Savart and others
π» β π· = π
π» β π΅ = 0
π» Γ πΈ = βππ©
ππ‘
π» Γ π» = π +ππ«
ππ‘
π· = π0 πΈ, π΅ = π0π», π0π0π2 = 1 In vacuum
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1st Maxwell equation
Equivalent to Gaussβs Flux Theorem, or Coulombβs Law
π» β πΈ =π
π0 β π» β πΈππ = πΈ
ππ
dπ =1
π0 πππ
π
=π
π0
The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface Field generated by a point charge:
πΈ =π
4ππ0
π
π3 β πΈπ =
π
4ππ0
1
π2
πΈ
π πβπππ
dπ =π
4ππ0
ππ
π2π πβπππ
=π
π0
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The net magnetic flux out of any closed surface is zero. Surround a magnetic dipole with a closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole. If there were a magnetic monopole source, this would give a non-zero integral.
π» β π΅ = 0
2nd Maxwell equation
π΅ππ = 0
Gauss Law for Magnetism
The magnetic field can be derived from a vector potential A defined by B = βΓA.
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 7
3rd Maxwell equation
Relates electric field and non constant magnetic field Equivalent to Faradayβs law of Induction
π» Γ πΈ = βππ©
ππ‘
π» Γ πΈ β ππ = β ππ©
ππ‘ππ
ππ
πΈ β ππ = βπ
ππ‘ π΅ β ππ = β
πΞ¦
ππ‘ππΆ
Faradayβs Law is the basis for electric generators, inductors and transformers.
The electromotive force round a circuit is proportional to the rate of change of flux of magnetic field through the circuit.
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 8
4th Maxwell equation
π» Γ π» = π +ππ«
ππ‘ β π» Γ
1
ππ΅ = ππ π +
1
π2π ππΈ
ππ‘
From Ampereβs (Circuital) Law : π» Γ π΅ = π0π
Relates magnetic field and not constant electric field. Equivalent to Faradayβs law of Induction
π΅ β ππ = π» Γ π΅ β ππ = π0 π β ππ = π0πΌ
πππΆ
Satisfied by the field for a steady line current (Biot-Savart Law, 1820):
Type equation here.
π΅ =π0πΌ
4π ππ Γ π
π3
Which for a straight line current π΅ =π0πΌ
2ππ
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Example
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 10
Example: Field in a permeable dipole
β’ Cross section of dipole magnet
10
g
Integration loop
enclosed
gap
gap
steelC
IgB
gBldBldH
0
0steelin path
1
g
INB turns
gap0
msteel mgap
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 11
Relativity
For the most part, we will use SI units, except
β Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J]
β Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2]
β Momentum: eV/c [proton @ b=.9 = 1.94 GeV/c]
2222
2
2
2
energy kinetic
energy total
momentum
1
1
pcmcE
mcEK
mcE
mvp
E
pc
c
v
b
b
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 12
Incremental relationships between
energy, velocity and momentum
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Energy (MeV)
b
electrons
protons
0 5 100
0.2
0.4
0.6
0.8
1
Energy (GeV)
b
electrons
protons
Particle velocity as a function of kinetic energy
Electrons are ultrarelativistic at few MeV
Protons at few GeV (mass 2000 times electron mass)
1 bbc
vParticle at light velocity c
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Electrodynamic Potentials
We can write the electric and magnetic fields in terms of Vector and
Scalar potentials
14
t
AE
trAB
,
correctically relativist ;
for ;
dt
pd
cvdt
vdm
dt
pdBvEeF
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Lorentzβs Force
Particle dynamics are governed by the Lorentz force law
vdt
cvdt
vdmF
BvEqF
any for
for
field electric E
field magneticB
Acceleration Bending and focusing
Beam
Electric field
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Particle in a magnetic field
The magnetic force always acts at right angles to
the charge motion, the magnetic force can do no
work on the charge. The B-field cannot speed up
or slow down a moving charge; it can only change
the direction in which the charge is moving.
The component of velocity of the charged particle that is parallel to the magnetic field is unaffected, i.e. the charge moves at a constant speed along the direction of the magnetic field. Negatively charged particles circulate in the opposite direction as positively charged particles. The direction can be found using the right hand rule applied to the perpendicular component of the velocity
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Motion in constant electric field
π πππΎπ£
ππ‘= q πΈ
πΎπ£ =ππΈ
ππt
πΎ2 = 1 +πΎπ£
π
2
πΎ = 1 +ππΈπ‘
πππ
2
If the electric field is constant:
πΈ = πΈ, 0,0
It can be demonstrated that
πππ2 πΎ β 1 = ππΈ(π₯ β π₯π)
Uniform acceleration in straight line
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Motion in a constant magnetic field
β’ A charged particle moves in circular arcs
of radius r with angular velocity w:
m
qBv
mvqvB
vmdt
dmBvqv
Bvqdt
vdmvm
dt
d
dt
rw
r
wr
wrw
2
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
Example: Cyclotron (1930βs)
β’ A charged particle in a uniform magnetic field will follow a circular path of radius
side view
B
r
top view
B
m
qBf
m
qB
vf
cvqB
mv
s
r
r
2
)!(constant! 2
2
)(
MHz ][2.15 TBfC
βCyclotron Frequencyβ
For a proton:
Accelerating βDEESβ
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Rigidity
Relation between radius and
momentum
π΅π = π
π
How hard (or easy) is a particle to deflect? β’ Often expressed in [T-m] (easy to calculate B) β’ Be careful when qβ e!!
π΅π[ππ] β 3.33π[πΊπππ]
π[π]
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
Electrons and protons (now in MeV)
πΈπ = 0.511 πππ ππ 938.27 πππ πΈπ‘ππ‘ = πΈπππ + πΈπ
πΎ =πΈπ‘ππ‘πΈπ, π½ = 1 β
1
πΎ2
π = π½πΈπ‘ππ‘
π΅π = 3.33 10β3π
Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
πΈπ = 0.511 πππ πΈπ = 938.27 πππ πΈπ = 939.57πππ ππ, π, π β π΄ = π΄π‘ππππ πππ π π + π
πΈπ = ππΈπ + ππΈπ + πππΈπ β π΄ β 0.8
πΈπ‘ππ‘ = π΄ πΈπππ + πΈπ
πΎ =πΈπ‘ππ‘πΈπ, π½ = 1 β
1
πΎ2
π = π½πΈπ‘ππ‘
π΅π = 3.33 10β3π/Q
Ions Kinetic energy per nucleon: Ekin
Total charge = Qe
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Example: particle spectrometer
Identify particle momentum by measuring bending angle from a calibrated magnetic field B
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Simplified Particle Motion
Design trajectory
β’ Particle motion will be expanded around a design trajectory or orbit
β’ This orbit can be over linacs, transfer lines, rings
Separation of fields: Lorentz force
β’ Magnetic fields from static or slowly-changing magnets transverse to design trajectory
β’ Electric fields from high-frequency RF cavities in direction of design trajectory
β’ Relativistic charged particle velocities
Slide from T. Satogata / January 2015 USPAS Accelerator Physics
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Others
Specially: Calcul
Matrix formalism Differential equations
Electromagnetism Special relativity
Waves and Optics
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References
β’ Particle Accelerator Physics β Helmut Wiedemann, Third
Edition, Springer
β’ Cern Accelerator School (CAS)
(http://cas.web.cern.ch/cas/) , including links to other
schools
β’ U.S. Particle Accelerator School (USPAS)
(http://uspas.fnal.gov/)
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Exercises
β’ Find the velocity in terms of c of an
electron at 3 GeV (ALBA), of a proton at
250 MeV (CNAO) and at 6.5 TeV (LHC)
β’ Derive the expression of the relationship
between momentum and energy change ππ
π=1
π½2ππΈ
πΈ