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Accelerator Physics Accelerator Physics Contents: 1. Introduction __________________________________________________________________ 3 2. Magnets _____________________________________________________________________ 4 2.1. General remarks on the calculation of magnetic fields ____________________________________ 4 2.2. Particle Beam Guidance ____________________________________________________________ 8 2.3. Particle Beam Focusing ____________________________________________________________ 10 2.4. Correction of Chromatic Errors ______________________________________________________ 14 2.5. Multipole Field Expansion __________________________________________________________ 17 2.6. Effective Field length ______________________________________________________________ 21 3. Linear Beam Optics ___________________________________________________________ 22 3.1. Equations of Motion in a moving Reference System _____________________________________ 22 3.2. Matrix Formalism _________________________________________________________________ 27 3.2.1. Drift Space ___________________________________________________________________________28 3.2.2. Dipole Magnets _______________________________________________________________________29 3.2.3. Quadrupole Magnets ___________________________________________________________________39 3.2.4. Particle Orbits in a System of Magnets _____________________________________________________41 3.3. Particle Beams and Phase Space _____________________________________________________ 43 3.3.1. Beam Emittance _______________________________________________________________________43 3.3.2. Twiss Parameters ______________________________________________________________________47 Terascale Accelerator School 2008 1 W. Hillert Accelerator Physics 3.3.3. Betatron Functions _____________________________________________________________________50 3.3.4. Transformation in Phase Space ___________________________________________________________53 3.3.6. Dispersion Functions ___________________________________________________________________57 3.3.7. Path Length and Momentum Compaction __________________________________________________59 4. Circular Accelerators __________________________________________________________ 61 4.1. Periodic Focusing Systems __________________________________________________________ 61 4.1.1. Periodic Betatron Functions ______________________________________________________________61 4.1.2. Stability Criterion ______________________________________________________________________63 4.1.3. General FODO Lattice ___________________________________________________________________65 4.2. Transverse Beam Dynamics_________________________________________________________ 68 4.2.1. Betatron Tune _________________________________________________________________________68 4.2.2. Filamentation _________________________________________________________________________69 4.2.3. Normalized Coordinates _________________________________________________________________70 4.2.4. Closed orbit distortions _________________________________________________________________71 4.2.5. Gradient Errors ________________________________________________________________________75 4.2.6. Optical Resonances ____________________________________________________________________78 4.2.7. Chromaticity __________________________________________________________________________81 4.3. Longitudinal Beam Dynamics _______________________________________________________ 86 4.3.1. Equation of Motion in Phase Space ________________________________________________________86 4.3.2. Small Oscillation Amplitudes _____________________________________________________________89 4.3.3. Large Amplitude Oscillations _____________________________________________________________93 Terascale Accelerator School 2008 2 W. Hillert Accelerator Physics 1. Introduction Particles in electric and magnetic fields: Lorentz-force F = eE⋅+×( v B) Energy increase is related to longitudinal electric fields: ⎛⎞ F iids= e⋅+×=⋅=⋅⎜⎟ E dsv B i ds e E ds e U ∫∫∫⎝⎠( ) ∫ Beam deflection due to perpendicular fields, has to cancel centrifugal force: ! v2 FmceEcBm==⋅+×=γβ β γ ⊥⊥⊥00()R Electric stiffness and magnetic stiffness: 1 epeR B == R E ⊥ v ⊥ Ultrarelativistic particles: v ≈ c ⇒⇔RB⊥ =1Tm RE⊥ ≈ 300 MV m Terascale Accelerator School 2008 3 W. Hillert Accelerator Physics 2. Magnets 2.1. General remarks on the calculation of magnetic fields The calculation of magnetic fields is based on Maxwell’s equation ∇ ×=H j . We are interested in the magnetic field close to the particles orbit where j = 0 , thus yielding ∇ ×=H 0 . H may therefore be expressed in terms of a scalar potential ϕ by H = −∇Φ , giving ΔΦ=0 . The magnetic field usually is generated by an electrical current I in current carrying coils surrounding magnet poles made of ferromagnetic material. A ferromagnetic re- turn yoke surrounds the excitation coils providing an efficient return path for the magnetic flux. Terascale Accelerator School 2008 4 W. Hillert Accelerator Physics The magnetic potential Φ is defined by its equipotential lines in the transverse plane and therefore by the surface of the ferromagnetic poles. Due to the relation B = μμ0 r H , the magnetic field in the gap of a magnet may be calculated from a closed loop integral nI⋅= Hds ⋅ = H ⋅ ds + H , ∫ ∫∫0 E gap yoke Terascale Accelerator School 2008 5 W. Hillert Accelerator Physics where n is the number of windings and I the current of the excitation coil of the mag- net. At the surface of the magnetic poles, the continuity relation gives 1 HHEE=⇒00 HH . μr Assuming a very large permeability μr , the second integral term may be neglected. We finally obtain for the magnetic field inside the gap B ⋅ ds=⋅⋅μ n I ∫ 0 gap Concentrating on the major types of magnets (so called upright magnets) mainly used in particles accelerators and beam transport lines, it is sufficient to define the vertical component Gxz () of the magnetic field along the horizontal axis of the magnet (where we put z = 0 ) and to assume a constant field distribution along the longitudin- al axis (∂Bs∂=0 ). This leads to the set-up Terascale Accelerator School 2008 6 W. Hillert Accelerator Physics Bzz(,)xz= G () x + f ()z , and with B = −∇Φ , one has Φ(,)xz=− B ⋅ dz =− G () x ⋅ z −f ()zdz ⋅ , ∫ zz∫ and therefore dG2 () x dfz () ΔΦ=−z z − =0 dx2 dz dG22() x 1 dG () x ⇒=−⋅=−f ()zzdzzzz2 ∫ dx222 dx Finally, we obtain for the scalar potential of the magnetic field: 1()dG2 x Φ(,)xz=− G () x ⋅ z +z ⋅ z3 z 6 dx2 Terascale Accelerator School 2008 7 W. Hillert Accelerator Physics 2.2. Particle Beam Guidance Guide fields are used to deflect particles along a predefined path. These fields, gener- ated by Dipole Magnets have to be homogenous, leading to the set-up dG2 () x Gx( )= B=⇒ const.z = 0 z 0 dx2 The corresponding magnetic potential follows to Φ(,)xz=− B0 ⋅ z, defining the profile of the magnet poles to flat parallel poles at a distance h: Terascale Accelerator School 2008 8 W. Hillert Accelerator Physics Typical types of dipole magnets are the C-Magnet, the H-Magnet and the Window- Frame Magnet: C-Magnet H-Magnet Window-Frame Magnet The magnetic field inside the gap is related to the current of the coils by nI⋅ B = μ 00h The strength of a dipole magnet is usually normalized to the particles momentum, giving the inverse bending radius 1 R or the curvature κ : 1 eenIμ ⋅ κ ==B = 0 R pph0 Terascale Accelerator School 2008 9 W. Hillert Accelerator Physics 2.3. Particle Beam Focusing Focusing fields are used to keep the particle beam together and to generate specifical- ly desired beam properties at selected points along a beam transport line. We need a magnetic field which increases linearly with increasing distance from the axis, leading to the set-up: ∂ B Gx()=⋅ gx with g = z z ∂ x generated by Quadrupole Magnets. The corresponding potential follows to Φ(,)xz =−g ⋅xz ⋅ , defining the profile of the magnet poles to four hyperbolic poles Φ a2 zx()=±0 =± g ⋅ xx2 at a distance a =Φ2 0 g from the axis. Terascale Accelerator School 2008 10 W. Hillert Accelerator Physics The magnetic field inside the gap is determined from the gradient of the magnetic ∂Φ ∂Φ potential, giving: Bg= −=⋅zBand =−=⋅g x xz∂∂xz The “restoring” force acting on the particles is F =⋅eBe(vv ×) =g ⋅()xeˆˆx − zez A quadrupole magnet is therefore focusing only in one plane and defocusing in the other; depending on the sign of g. Terascale Accelerator School 2008 11 W. Hillert Accelerator Physics The g-parameter may be related to the current of the coils by evaluating the closed 12 01 loop integral n⋅=⋅= I H ds H ⋅+ ds H ⋅+ ds H ⋅≈ ds H ⋅ ds, ∫∫∫∫∫000E 01 20 g 2⋅ μ0 ⋅⋅nI One obtains with Hds⋅= rdr ⋅: g = 2 μ0 a Again we would like to normalize g to the particles momentum, giving the quadru- eenI2 μ ⋅ pole strength k: kg== 0 ppa2 Terascale Accelerator School 2008 12 W. Hillert Accelerator Physics The focal length of a thin quadrupole magnet of length L can be derived from the def- lection angle α of the particles beam and its relation to the quadrupole strength k, x tanα = f LLe tanα == =g xL = xkL RpeBp()z 1 giving: = kL⋅ f Here we have assumed the length L to be short compared to the focal length f such that R does not change significantly within the quadrupole magnetic field. Terascale Accelerator School 2008 13 W. Hillert Accelerator Physics 2.4. Correction of Chromatic Errors To correct focusing errors of the quadrupole magnets we need the nonlinear magnetic fields of Sextupole Magnets, in which the field
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