Basic Maths and Physics for Accelerators
Transcript of Basic Maths and Physics for Accelerators
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
C. Biscari
Basic Maths and Physics for
Accelerators
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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
Avogadroโs number A 6.021415 1023 1/mol
Classical electron radius rc 2.8179433 10-15 m
Planckโs constant h 4.1356675 10-15 eV s
l of 1 eV photon ฤงc/e 12398.419 ศฆ
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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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 3
Vector calculus
Gradient of a scalar function
๐ ๐ฅ, ๐ฆ, ๐ง, ๐ก
๐ป๐ =๐๐
๐๐ฅ,๐๐
๐๐ฆ,๐๐
๐๐ง
Divergence
๐ป โ ๐น =๐๐น1
๐๐ฅ+๐๐น2
๐๐ฆ+๐๐น3
๐๐ง
For a vector
๐น = ๐น1, ๐น2, ๐น3
Curl
๐ป ร ๐น
=๐๐น3๐๐ฆ
โ๐๐น2๐๐ง,๐๐น1๐๐ง
โ๐๐น3๐๐ฅ,๐๐น2๐๐ฅโ๐๐น1๐๐ฆ
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
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 4
Maxwellโs 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
๐ป โ ๐ท = ๐
๐ป โ ๐ต = 0
๐ป ร ๐ธ = โ๐๐ฉ
๐๐ก
๐ป ร ๐ป = ๐ +๐๐ซ
๐๐ก
๐ท = ๐0 ๐ธ, ๐ต = ๐0๐ป, ๐0๐0๐2 = 1 In vacuum
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 5
1st Maxwell equation
Equivalent to Gaussโs Flux Theorem, or Coulombโs Law
๐ป โ ๐ธ =๐
๐0 โ ๐ป โ ๐ธ๐๐ = ๐ธ
๐๐
d๐ =1
๐0 ๐๐๐
๐
=๐
๐0
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:
๐ธ =๐
4๐๐0
๐
๐3 โ ๐ธ๐ =
๐
4๐๐0
1
๐2
๐ธ
๐ ๐โ๐๐๐
d๐ =๐
4๐๐0
๐๐
๐2๐ ๐โ๐๐๐
=๐
๐0
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 6
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.
๐ป โ ๐ต = 0
2nd Maxwell equation
๐ต๐๐ = 0
Gauss Law for Magnetism
The magnetic field can be derived from a vector potential A defined by B = โรA.
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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 Faradayโs law of Induction
๐ป ร ๐ธ = โ๐๐ฉ
๐๐ก
๐ป ร ๐ธ โ ๐๐ = โ ๐๐ฉ
๐๐ก๐๐
๐๐
๐ธ โ ๐๐ = โ๐
๐๐ก ๐ต โ ๐๐ = โ
๐ฮฆ
๐๐ก๐๐ถ
Faradayโs 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.
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 8
4th Maxwell equation
๐ป ร ๐ป = ๐ +๐๐ซ
๐๐ก โ ๐ป ร
1
๐๐ต = ๐๐ ๐ +
1
๐2๐ ๐๐ธ
๐๐ก
From Ampereโs (Circuital) Law : ๐ป ร ๐ต = ๐0๐
Relates magnetic field and not constant electric field. Equivalent to Faradayโs law of Induction
๐ต โ ๐๐ = ๐ป ร ๐ต โ ๐๐ = ๐0 ๐ โ ๐๐ = ๐0๐ผ
๐๐๐ถ
Satisfied by the field for a steady line current (Biot-Savart Law, 1820):
Type equation here.
๐ต =๐0๐ผ
4๐ ๐๐ ร ๐
๐3
Which for a straight line current ๐ต =๐0๐ผ
2๐๐
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 9
Example
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Example: Field in a permeable dipole
โข 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
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 11
Relativity
For the most part, we will use SI units, except
โ Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J]
โ Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2]
โ 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
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 12
Incremental relationships between
energy, velocity and momentum
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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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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 14
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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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 15
Lorentzโs 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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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 16
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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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 17
Motion in constant electric field
๐ ๐๐๐พ๐ฃ
๐๐ก= q ๐ธ
๐พ๐ฃ =๐๐ธ
๐๐t
๐พ2 = 1 +๐พ๐ฃ
๐
2
๐พ = 1 +๐๐ธ๐ก
๐๐๐
2
If the electric field is constant:
๐ธ = ๐ธ, 0,0
It can be demonstrated that
๐๐๐2 ๐พ โ 1 = ๐๐ธ(๐ฅ โ ๐ฅ๐)
Uniform acceleration in straight line
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 18
Motion in a constant magnetic field
โข 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
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
Example: Cyclotron (1930โs)
โข 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
โCyclotron Frequencyโ
For a proton:
Accelerating โDEESโ
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 20
Rigidity
Relation between radius and
momentum
๐ต๐ = ๐
๐
How hard (or easy) is a particle to deflect? โข Often expressed in [T-m] (easy to calculate B) โข Be careful when qโ e!!
๐ต๐[๐๐] โ 3.33๐[๐บ๐๐๐]
๐[๐]
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
Electrons and protons (now in MeV)
๐ธ๐ = 0.511 ๐๐๐ ๐๐ 938.27 ๐๐๐ ๐ธ๐ก๐๐ก = ๐ธ๐๐๐ + ๐ธ๐
๐พ =๐ธ๐ก๐๐ก๐ธ๐, ๐ฝ = 1 โ
1
๐พ2
๐ = ๐ฝ๐ธ๐ก๐๐ก
๐ต๐ = 3.33 10โ3๐
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics
๐ธ๐ = 0.511 ๐๐๐ ๐ธ๐ = 938.27 ๐๐๐ ๐ธ๐ = 939.57๐๐๐ ๐๐, ๐, ๐ โ ๐ด = ๐ด๐ก๐๐๐๐ ๐๐๐ ๐ ๐ + ๐
๐ธ๐ = ๐๐ธ๐ + ๐๐ธ๐ + ๐๐๐ธ๐ โ ๐ด โ 0.8
๐ธ๐ก๐๐ก = ๐ด ๐ธ๐๐๐ + ๐ธ๐
๐พ =๐ธ๐ก๐๐ก๐ธ๐, ๐ฝ = 1 โ
1
๐พ2
๐ = ๐ฝ๐ธ๐ก๐๐ก
๐ต๐ = 3.33 10โ3๐/Q
Ions Kinetic energy per nucleon: Ekin
Total charge = Qe
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 23
Example: particle spectrometer
Identify particle momentum by measuring bending angle from a calibrated magnetic field B
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 24
Simplified Particle Motion
Design trajectory
โข Particle motion will be expanded around a design trajectory or orbit
โข This orbit can be over linacs, transfer lines, rings
Separation of fields: Lorentz force
โข Magnetic fields from static or slowly-changing magnets transverse to design trajectory
โข Electric fields from high-frequency RF cavities in direction of design trajectory
โข Relativistic charged particle velocities
Slide from T. Satogata / January 2015 USPAS Accelerator Physics
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 25
Others
Specially: Calcul
Matrix formalism Differential equations
Electromagnetism Special relativity
Waves and Optics
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 26
References
โข Particle Accelerator Physics โ Helmut Wiedemann, Third
Edition, Springer
โข Cern Accelerator School (CAS)
(http://cas.web.cern.ch/cas/) , including links to other
schools
โข U.S. Particle Accelerator School (USPAS)
(http://uspas.fnal.gov/)
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Accelerator Physics - UAB 2015-16 C. Biscari - Lecture 2 Basic Maths & Physics 27
Exercises
โข 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)
โข Derive the expression of the relationship
between momentum and energy change ๐๐
๐=1
๐ฝ2๐๐ธ
๐ธ