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Relativity, EM Forces - Historical Introduction Accelerator Physics Alex Bogacz (Jefferson Lab) / [email protected] Geoff Krafft (Jefferson Lab/ODU) / [email protected] Timofey Zolkin (U. Chicago/Fermilab) / [email protected] Thomas Jefferson National Accelerator Facility Lecture 1Accelerator− Relativity, Physics EM Forces, Intro Operated by JSA for the U.S. Department of Energy USPAS, Fort Collins, CO, June 10-21, 2013 1 Introductions and Outline Syllabus Week 1 Week 2 Introduction Course logistics, Homework, Exam: http://casa/publications/USPAS_Summer_2013.shtml Relativistic mechanics review Relativistic E&M review, Cyclotrons Survey of accelerators and accelerator concepts Thomas Jefferson National Accelerator Facility Lecture 1Accelerator− Relativity, Physics EM Forces, Intro Operated by JSA for the U.S. Department of Energy 2 Syllabus – week 1 Mon 10 June 0900-1200 Lecture 1 ‘Relativity, EM Forces - Historical Introduction Mon 10 June 1330-1630 Lecture 2 ‘Weak focusing and Transverse Stability’ Tue 11 June 0900-1200 Lecture 3 ‘Linear Optics’ Tue 11 June 1330-1630 Lecture 4 ‘Phase Stability, Synchrotron Motion’ Wed 12 June 0900-1200 Lecture 5 ‘Magnetic Multipoles, Magnet Design’ Wed 12 June 1330-1630 Lecture 6 ‘Particle Acceleration’ Thu 13 June 0900-1200 Lecture 7 ‘Coupled Betatron Motion I’ Thu 13 June 1330-1630 Lecture 8 ‘Synchrotron Radiation’ Fri 14 June 0900-1200 Lecture 9 ‘Coupled Betatron Motion II’ Fri 14 June 1330-1530 Lecture 10 ‘Radiation Distributions’ Thomas Jefferson National Accelerator Facility Lecture 1Accelerator− Relativity, Physics EM Forces, Intro Operated by JSA for the U.S. Department of Energy 3 Syllabus – week 2 Mon 17 June 0900-1200 Lecture 11 ‘Nonlinear Dynamics, Resonance Theory’ Mon 17 June 1330-1630 Lecture 12 ‘X-ray Sources/ FELs’ Tue 18 June 0900-1200 Lecture 13 ‘Nonlinear Dynamics, Chaos’ Tue 18 June 1330-1630 Lecture 14 ‘Statistical Effects’ Wed 19 June 0900-1200 Lecture 15 ‘Cooling Theory’ Wed 19 June 1330-1630 Lecture 16 ‘Radiation Damping’ Thu 20 June 0900-1200 Lecture 17 ‘Low Emittance Lattices’ Thu 20 June 1330-1630 Lecture 18 ‘Feedback Systems’ Thomas Jefferson National Accelerator Facility Lecture 1Accelerator− Relativity, Physics EM Forces, Intro Operated by JSA for the U.S. Department of Energy 4 Homework and Schedule Homework is more than 1/3 of your grade (35%) Tim is grading Collected at start of every morning class Tim’s homework is to get it back to you the next day Lectures will run 09:00-12:00, 13:30-16:30 Exams Mid-term (Friday, June 14) Final (Friday, June 21) Thomas Jefferson National Accelerator Facility Lecture 1Accelerator− Relativity, Physics EM Forces, Intro Operated by JSA for the U.S. Department of Energy 5 Relativity, EM Forces: Historical Introduction Alex Bogacz, Geoff Krafft and Timofey Zolkin Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy USPAS, Fort Collins, CO, June 10-21, 2013 6 Relativity Review Accelerators: applied special relativity Relativistic parameters: Later β and γ will also be used for other quantities, but the context should usually make them clear γ=1 (classical mechanics) to ~2.05 x105 (to date) Total energy U, momentum p, and kinetic energy W Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 7 Convenient Units How much is a TeV? Energy to raise 1g about 16 μm against gravity Energy to power 100W light bulb 1.6 ns But many accelerators have 1010-12 particles Single bunch “instantaneous power” of tens of Terawatts Highest energy cosmic ray ~300 EeV (3x1020 eV or 3x108 TeV!) Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 8 Relativity Review (Again) Accelerators: applied special relativity Relativistic parameters: Later β and γ will also be used for other quantities, but the context should usually make them clear γ=1 (classical mechanics) to ~2.05 x105 Total energy U, momentum p, and kinetic energy W Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 9 Convenient Relativity Relations All derived in the text, hold for all γ In “ultra” relativistic limit β≈1 Usually must be careful below γ ≈ 5 or U ≈ 5 mc2 Many accelerator physics phenomena scale with γk or (βγ)k Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 10 Frames and Lorentz Transformations The lab frame will dominate most of our discussions But not always (synchrotron radiation, space charge…) Invariance of space-time interval (Minkowski) Lorentz transformation of four-vectors For example, time/space coordinates in z velocity boost Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 11 Four-Velocity and Four-Momentum The proper time interval dτ=dt/γ is Lorentz invariant So we can make a velocity 4-vector We can also make a 4-momentum Double-check that Minkowski norms are invariant Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 12 Mandelstam Variables Lorentz-invariant two-body kinematic variables p1-4 are four-momenta √s is the total available center of mass energy Often quoted for colliders Used in calculations of other two-body scattering processes Moller scattering (e-e), Compton scattering (e-γ) Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 13 Relativistic Newton Equation But now we can define a four-vector force in terms of four- momenta and proper time: We are primarily concerned with electrodynamics so now we must make the classical electromagnetic Lorentz force obey Lorentz transformations Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 14 Relativistic Electromagnetism Classical electromagnetic potentials can be shown to combine to a four-potential (with c = 1): The field-strength tensor is related to the four-potential E/B fields Lorentz transform with factors of γ, (βγ) Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 15 Lorentz Lie Group Generators Lorentz transformations can be described by a Lie group where a general Lorentz transformation is where L is 4x4, real, and traceless. With metric g, the matrix gL is also antisymmetric, so L has the general six-parameter form Deep and profound connection to EM tensor Fαβ J.D. Jackson, Classical Electrodynamics 2nd Ed, Section 11.7 Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 16 Relativistic Electromagnetism II The relativistic electromagnetic force equation becomes Thankfully we can write this in somewhat simpler terms That is, “classical” E&M force equations hold if we treat the momentum as relativistic, If we dot in the velocity, we get energy transfer Unsurprisingly, we can only get energy changes from electric fields, not (conservative) magnetic fields Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 17 Constant Magnetic Field (E = 0) In a constant magnetic field, charged particles move in circular arcs of radius ρ with constant angular velocity ω: For we then have Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 18 Rigidity: Bending Radius vs Momentum Accelerator Beam (magnets, geometry) This is such a useful expression in accelerator physics that it has its own name: rigidity Ratio of momentum to charge How hard (or easy) is a particle to deflect? Often expressed in [T-m] (easy to calculate B) Be careful when q≠e!! A very useful expression Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 19 Cyclotron Frequency Another very useful expression for particle angular frequency in a constant field: cyclotron frequency In the nonrelativistic approximation Revolution frequency is independent of radius or energy! Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 20 Lawrence and the Cyclotron Can we repeatedly spiral and accelerate particles through the same potential gap? Accelerating gap ΔΦ Ernest Orlando Lawrence Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 21 Cyclotron Frequency Again Recall that for a constant B field Radius/circumference of orbit scale with velocity Circulation time (and frequency) are independent of v Apply AC electric field in the gap at frequency frf Particles accelerate until they drop out of resonance Note a first appearance of “bunches”, not DC beam Works best with heavy particles (hadrons, not electrons) Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 22 A Patentable Idea 1934 patent 1948384 Two accelerating gaps per turn! Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S. Department of Energy 23 All The Fundamentals of an Accelerator Large static magnetic fields for guiding (~1T) ~13 cm But no vertical focusing HV RF electric fields for accelerating (No phase focusing) (Precise f control) p/H source, injection, extraction, vacuum 13 cm: 80 keV 28 cm: 1 MeV 69 cm: ~5 MeV … 223 cm: ~55 MeV (Berkeley) Thomas Jefferson National Accelerator Facility Lecture 1 − Relativity, EM Forces, Intro Operated by JSA for the U.S.
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