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Introduction to Accelerators Introduction to Accelerators Lecture 4 Basic Properties of Particle Beams William A. Barletta Director, United States Particle Accelerator School Dept. of Physics, MIT US Particle Accelerator School Homework item US Particle Accelerator School From the last lecture US Particle Accelerator School We computed the B-field from current loop with I = constant By the Biot-Savart law we found that on the z-axis I 2 2IR2 B = Rsin d zˆ = zˆ cr2 2 2 3/2 0 cR()+ z What happens if we drive the current to have a time variation? r R US Particle Accelerator School The far field B-field has a static dipole form Importantly the ring of current does not radiate US Particle Accelerator School Question to ponder: What is the field from this situation? r R We’ll return to this question in the second half of the course US Particle Accelerator School Is this really paradoxical? Let’s look at Maxwell’s equations Take the curl of xE Hence US Particle Accelerator School The dipole radiation field: note the similarity to the static dipole US Particle Accelerator School Now on to beams US Particle Accelerator School Beams: particle bunches with directed velocity Ions - either missing electrons (+) or with extra electrons (-) Electrons or positrons Plasma - ions plus electrons Source techniques depend on type of beam & on application US Particle Accelerator School Electron sources - thermionic Heated metals Some electrons have energies above potential barrier Cannot escape + HV Enough energy to escape # of electrons Work function = Electrons in a metal obey Fermi statistics dn(E) 1 = AE (EEF )/kT dE []e +1 US Particle Accelerator School Electrons with enough momentum can escape the metal Integrating over electrons going in the z direction with 2 pz /2m > E F + yields 3 Je = dpx dpy dpx (2 /h ) f (E)vz pz, free some considerable manipulation yields the Richardson-Dushman equation 2 q I AT exp kBT A=1202 mA/mm2K2 US Particle Accelerator School Brightness of a beam source A figure of merit for the performance of a beam source is the brightness Beam current Emissivity (J) B= = Beam area o Beam Divergence Temperature/mass Je Je = 2 = kT kT 2 2 moc moc Typically the normalized brightness is quoted for = 1 US Particle Accelerator School Other ways to get electrons over the potential barrier Field emission Sharp needle enhances electric field + HV Photoemission from metals & semi-conductors Photon energy exceeds the work function These sources produce beams with high current densities and low thermal energy This is a topic for a separate lecture US Particle Accelerator School Anatomy of an ion source Container Extraction for plasma electrodes Beam Energy in to ionize gas out Gas in Electron filter Plasma Electron beams can also be used to ionize the gas or sputter ions from a solid US Particle Accelerator School What properties characterize particle beams? US Particle Accelerator School Beams have directed energy The beam momentum refers to the average value of pz of the particles pbeam = <pz> The beam energy refers to the mean value of 2 1/2 E = p c 2 + m2c 4 beam []z For highly relativistic beams pc>>mc2, therefore Ebeam = pz c US Particle Accelerator School Measuring beam energy & energy spread Magnetic spectrometer - for good resolution, p one needs small sample emittance , (parallel particle velocities) a large beamwidth w in the bending magnet a large angle p = w US Particle Accelerator School Beam carry a current I ~ ne<vz> pulse Duty factor = T pulse Q I peak = Q pulse Qtot I ave = T Imacro I US Particle Accelerator School Measuring the beam current Return current Beam current eam pipe Voltage meter Beam current Stripline waveguide Examples: Non-intercepting: Wall current monitors, waveguide pick-ups Intercepting: Collect the charge; let it drain through a current meter • Faraday Cup US Particle Accelerator School Collecting the charge: Right & wrong ways The Faraday cup Simple collector Proper Faraday cup US Particle Accelerator School Bunch dimensions Y For uniform charge distributions We may use “hard edge values For gaussian charge distributions y Use rms values x, y, z We will discuss measurements of X bunch size and charge distribution later x, , Z z US Particle Accelerator School But rms values can be misleading Gaussian beam Beam with halo We need to measure the particle distribution US Particle Accelerator School Measuring beam size & distribution US Particle Accelerator School Some other characteristics of beams Beams particles have random (thermal) motion 1/2 x px x = 2 > 0 pz Beams must be confined against thermal expansion during transport US Particle Accelerator School Beams have internal (self-forces) Space charge forces Like charges repel Like currents attract For a long thin beam 60 Ibeam (A) E sp (V /cm) = Rbeam (cm) Ibeam (A) B (gauss) = 5 Rbeam (cm) US Particle Accelerator School Net force due to transverse self-fields In vacuum: Beam’s transverse self-force scale as 1/2 Space charge repulsion: Esp, ~ Nbeam Pinch field: B ~ Ibeam ~ vz Nbeam ~ vz Esp 2 2 Fsp , = q (Esp, + vz x B) ~ (1-v ) Nbeam ~ Nbeam/ Beams in collision are not in vacuum (beam-beam effects) US Particle Accelerator School Example: Megagauss fields in linear collider electrons positrons At Interaction Point space charge cancels; currents add ==> strong beam-beam focus --> Luminosity enhancement --> Strong synchrotron radiation Consider 250 GeV beams with 1 kA focused to 100 nm Bpeak ~ 40 Mgauss US Particle Accelerator School Applications determine the desired beam characteristics Energy E = mc2 MeV to Te V Energy Spread (rms) E/E, ~0.1% Momentum spread / p/p 4 Beam current (peak ) I 10 – 10 A Pulse duration (FWH M ) Tp 50 fs - 50 ps Pulse length z mm - cm (Standard deviation) Charge per pulse Qb 1 nC # of Particles number N b Emittance is a Emittance (rms) 1 mm-mrad / measure of beam Normalized emittance n = quality Bunches per Mb 1- 100 macropul s e 7 Pulse repetition rate f 1 - 10 9 Effective bunch rate f Mb 1 - 10 US Particle Accelerator School What is this thing called beam quality? or How can one describe the dynamics of a bunch of particles? US Particle Accelerator School Coordinate space Each of Nb particles is tracked in ordinary 3-D space Orbit traces Not too helpful US Particle Accelerator School Configuration space: 6Nb-dimensional space for Nb particles; coordinates (xi, pi), i = 1,…, Nb The bunch is represented by a single point that moves in time Useful for Hamiltonian dynamics US Particle Accelerator School Configuration space example: 1 particle in an harmonic potential constant b px Fx = kx = mx˙˙ x But for many problems this description carries much more information than needed : We don’t care about each of 1010 individual particles But seeing both the x & px looks useful US Particle Accelerator School Option 3: Phase space (gas space in statistical mechanics) 6-dimensional space for Nb particles th The i particle has coordinates (xi, pi), i = x, y, z The bunch is represented by Nb points that move in time px x In most cases, the three planes are to very good approximation decoupled ==> One can study the particle evolution independently in each planes: US Particle Accelerator School Particles Systems & Ensembles The set of possible states for a system of N particles is referred as an ensemble in statistical mechanics. In the statistical approach, particles lose their individuality. Properties of the whole system are fully represented by particle density functions f6D and f2D : f6D ()x, px,y, py,z, pz dx dpx dy dpy dz dpz f2D ()xi, pi dxi dpi i =1, 2, 3 where f dx dp dy dp dz dp N 6D x y z = US Particle Accelerator School From: Sannibale USPAS lectures Longitudinal phase space In most accelerators the phase space planes are only weakly coupled. Treat the longitudinal plane independently from the transverse one Effects of weak coupling can be treated as a perturbation of the uncoupled solution In the longitudinal plane, electric fields accelerate the particles Use energy as longitudinal variable together with its canonical conjugate time Frequently, we use relative energy variation & relative time with respect to a reference particle E E 0 = =t t0 E 0 According to Liouville, in the presence of Hamiltonian forces, the area occupied by the beam in the longitudinal phase space is conserved US Particle Accelerator School From: Sannibale USPAS lectures Transverse phase space For transverse planes {x, px} and {y, py}, use a modified phase space where the momentum components are replaced by: w dx dy p p x = p y = WS PROJECTION xi ds yi ds w where s is the direction of motion s We can relate the old and new variables (for Bz 0) by dx dx p = m i = m v i = m c x i = x,y i 0 dt 0 s ds 0 i v 1 2 where = s and = ()1 2 c Note: xi and pi are canonical conjugate variables while x and xi’ are not, unless there is no acceleration ( and constant) US Particle Accelerator School From: Sannibale USPAS lectures Look again at our ensemble of harmonic oscillators px x Particles stay on their energy contour. Again the phase area of the ensemble is conserved US Particle Accelerator School Emittance describes the area in phase space of the ensemble of beam particles Emittance - Phase space volume of beam Phase space of an harmonic oscillator k(x) - frequency of rotation of a phase volume RMS emittance 2 R2(V 2 (R ) 2)/c 2 US Particle Accelerator School Twiss representation of the emittance A beam with arbitrary phase space distribution can be represented by an equivalent ellipse with area equal to the rms emittance divided by .
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