Particle Accelerators and Beam Optics

Home , Accelerator physics, Dipole magnet

Day 2

Mike Syphers Northern Illinois University Fermi National Accelerator Laboratory

Exotic Beams Summer School Argonne National Laboratory

July 2017 [email protected] Outline §Lecture 1 •Overview of types and uses of accelerators •Single-pass vs. repetitive systems •Transverse vs. longitudinal motion Questions? •Beams and particle distributions •Transverse beam optics

§Lecture 2 •Dispersion •Longitudinal beam dynamics »bunchers, re-bunchers; buckets and bunches •Optics modules •Accelerators for nuclear and high energy physics •Future directions

Summer 2017 MJS EBSS 2017 2 Quick Review B~ = B0y xˆ + B0x yˆ

A 1 T m @By B⇢ = · pu B = Q 300 MeV/c/u 0 x=0 ✓ ◆ @x |

2 2 x +2↵xx0 + x0 = ✏/⇡ x 10 x0 = 1 x’ x0 1 x0 ✓ ◆ ✓ F ◆✓ 0 ◆

x2 h i ✏ = ⇡ x2 x 2 xx 2 ⌘ ✏/⇡ h ih 0 ih 0i 2 p x0 ↵ h i K ⌘ ✏/⇡ ⌘ ↵ ✓ ◆ T xx0 Ki = Mi Ki 1 Mi ↵ h i ⌘ ✏/⇡ 1 1 xrms(s)= ✏(s)/⇡ ✏ xrms / p / pp p

Summer 2017 MJS EBSS 2017 3 Bending through Dipole Field

L B L L ✓ = = · ⇢ (B⇢) B~ q B L = · · ρ p

dθ

Summer 2017 MJS EBSS 2017 4 Dispersion p qB L B⇢ = ✓ = · q p

The bend angle (and/or dipole magnet: ✓ p p0 + p = focusing strength) depends ✓0 p upon momentum p0 p [i.e., in “opposite” at exit, to lowest order, x0 = ✓ Similar to index of 0 p direction of bend] refraction depending upon 1 p x `✓ frequency ⇡ 2 0 p likewise, for quadrupole: dipole steering “error” due x to a different momentum p0 + p —> “dispersion” p0 p f = f0(1 + ) focusing “error” due to a p different momentum Trajectory differences due to momentum differences referred to as “dispersion” —> “chromatic aberration" x(s) and, D(s, p/p) D(s) ⇡ ⌘ p/p “dispersion function”

Summer 2017 MJS EBSS 2017 5 Dispersion [2] (see E&S text for details…) Equation of Motion:

B0 1 1 B p p 1 1 p x + 0 + 0 x = x00 +( + 2 )x =0 becomes 00 2 B⇢ ⇢ 1+p/p B⇢ p0 + p · ⇢0(s) ⇢0 p0 + p ⇢✓ ◆

let x = D p/p, particular solution (must add the homogeneous solution, which we have found previously) betatron oscillation

then, p 1 B0 p0 p 1 p 1 p D00 + + D = p 1+p/p B⇢ p + p · ⇢ (s)2 p ⇢ p + p a driven betatron oscillation, 0 ⇢✓ ◆ 0 0 0 0 0 with a constant driving term. keep only terms linear in the relative momentum deviation, The “driver” is the dipole field p B0 1 p 1 p D00 + + D = within a bending magnet p B⇢ ⇢2 p ⇢ p 0 ✓ 0 ◆ 0 0 0 1 D00 + KD= ⇢0

so, solutions are plus sin pK` & cos pK` const.

Summer 2017 MJS EBSS 2017 6 1 D00 + KD= Dispersion [3] ⇢0

In terms of matrices… 1 s K = 0 : D00 = ,D0 = + D0 ⇢ ⇢ 0 1 s2 D = D + D0 s + in the limit of short, or “thin” 0 0 2 ⇢ elements, a bending magnet 1 2 D 1 s s /⇢ D0 primarily changes the slope of the 2 D0 = 01 s/⇢ D0 dispersion function by an amount 0 1 1 0 00 1 1 0 1 1 equal to the bend angle of the @ A @ A @ A magnet 1/! = 0 : same 2x2 as before D 0 D M 0 D0 = 0 D0 otherwise, the D transports roughly 0 1 1 0 01✓ ◆ 1 0 1 1 like a betatron oscillation @ A @ A @ A So, can use matrix methods (3x3 now; and 2x2 in “vertical” plane) to solve for: x, ↵x, x

y, ↵y, y

(& , if also have Dy,Dy0 Dx,Dx0 vertical bending)

Summer 2017 MJS EBSS 2017 7 FODO Channel §System of quadrupoles with alternating-sign gradients (F, D, F, …) separated by distance L, and with bending magnets in-between… x

p0 + dp0

p0 s ✓ ✓ ✓ ✓ ✓ F -F F -F F

§Can show …

2F 2✓ L D = 1 typically, on order max,min L ± 4F of a few meters ✓ ◆ Ex: D = 4 m, dp/p = 0.1%, then "x = 4 mm

Summer 2017 MJS EBSS 2017 8 Beam Size Including Dispersion

• Total excursion due to “off momentum” plus betatron oscillation:

x = x + D p/p ⌘ 2 2 2 2 x = x +2xD + D

• Assuming no correlation between x# and particle’s momentum:

x2 = x2 + D2 2 h i h i h i x2 = ✏/⇡ + D2 2 h i h i

Summer 2017 MJS EBSS 2017 9 Optical Modules

§Very often useful to think of optical systems in terms of modules §Each module has a purpose and/or special conditions to be met •general beam transport; achromatic; large dispersion for momentum selection, charge selection; small dispersion for isochronous transit; final focus onto target; long drift space for equipment; compact bending; etc. ... §Large/long systems are best generated with (stable!) periodic lens systems -- may or may not have bending §Often need longer spaces for instrumentation, RF, switching magnets, experiments, etc. §May need to match one focusing structure into a different focusing structure (e.g., change of cryomodule lengths, etc.)

§Simultaneously trying to control (α, β, D, D’)x,y [and sometimes ψx,y] as well as X,Y,Z and X’,Y’,Z’ of the ideal trajectory along the beam line! • various computer programs are good at this

Summer 2017 MJS EBSS 2017 10 Doublets and Triplets § FODO Cells • basic transport; equal spacing; easy analysis

betax § Doublets betay

• can be used to generate more space 20 15 10

§ Triplets Lattice Functions • can be used to keep beam “round” 5 0

0 2 4 6 8 10 12

Summer 2017 MJS EBSS 2017 11 Collins Straight Section Insertion §Wish to increase space between quadrupoles, perhaps to insert special element into the beam line. In order to match H and V optics simultaneously, we want αx = - αy and βx = βy at the match point(s). §Where can we use this?

Summer 2017 MJS EBSS 2017 12 Final Focus

Summer 2017 MJS EBSS 2017 13 Final Focus [2] 50 betax wall wall_2 betay sigx Dispx*10 beam direction §FRIB Final Focus: Dispy*10 sigy 150 §Beam size max/min 100 •max/min40 ~ 35

•max(linac)/min(target) ~ 10 Functions Lattice 50 30 0 6pi 0 1 2 3 4 5

s 50 sigx sigy 20 40 rms Beam Size (mm) rms Size Beam 30 6pi 20 10

rms Beam Size (mm) rms Size Beam 0.2pi

10 0.2pi 0

0 0 1 2 3 4 5

s 0 1 2 3 4 5 Summer 2017 MJS EBSS 2017 14 s Double-Bend Achromat

Summer 2017 MJS EBSS 2017 15 Achromatic Sections §FODO Achromatic Bend Section:

Summer 2017 MJS EBSS 2017 16 Chromatic Corrections and Chromaticity §Focusing effects from the magnets will also depend upon momentum: ′′ x + K(s, p)x =0 K = e(∂By(s)/∂x)/p §To give all particles the similar optics, regardless of momentum, need a “gradient” which depends upon momentum. Orbits spread out horizontally (or vertically) due to dispersion, can use a sextupole field:

• which gives (for y = 0) • • i.e., a ﬁeld gradient which depends upon momentum

§Chromaticity* is the variation of optics with momentum; use sextupole magnets to control/adjust; but, now introduces a nonlinear transverse field ... *In a synchrotron, “the” chromaticity is the variation of • can have a transverse dynamic aperture! the transverse oscillation frequency with momentum

Summer 2017 MJS EBSS 2017 17 In-Flight Production Example: NSCL’s CCF D.J. Morrissey, B.M. Sherrill, Philos. Trans. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 356 (1998) 1985. K500 Example: 86Kr 78Ni ion sources

coupling 86Kr14+, line 12 MeV/u K1200 A1900 focal plane p/p = 5% production transmission 86Kr34+, stripping target of 65% of the foil 140 MeV/u wedge produced 78Ni fragment yield after target fragment yield after wedge fragment yield at focal plane

Summer 2017 MJS EBSS 2017 18 Principle of Fragment Separator [2] Principle of Fragment SeparatorM. Hausmann,(2) FRIB ! Magnetic separation alone insufficient

! Numerous nuclides with similar A/Z

! But with different proton number (Z)

! Energy loss in matter is Z dependent ! Bethe formula (above)

! Interaction of beam with degrader (a piece of metal) leads to different velocity changes for different fragments ! Previously similar magnetic rigidities get “dispersed”

! This allows to separate these by magnetic rigidity ⟶ mass selection

Summer 2017 MJS EBSS 2017 19 Longitudinal Focusing

§sometimes referred to as “phase focusing” or “time focusing” §particles of different energy (momentum) move at different speeds, so tend to “spread out” relative to the “ideal” particle which is assumed to exist traveling with perfect synchronism with respect to the oscillating fields §wish to study the (longitudinal) motion of particles relative to this “synchronous particle”

Summer 2017 MJS EBSS 2017 20 Longitudinal Focusing

§time of flight — the “slip factor”

§Evolution due to dp/p or dW/W

§Longitudinal focusing, time of arrival: • bunchers, rebunchers, debunchers

Summer 2017 MJS EBSS 2017 21 Momentum Compaction Factor

§How does path length along the beam line depend upon momentum? • in straight sections, no difference; in bending regions, can be different

Look closely at an infinitesimal section along the ideal trajectory…

ds1 p ds ds1 x = D d✓ = = ds p ⇢ ⇢ + x ⇢ + x ⇢ d✓ ds ds = 1 ds 1 ⇢ ✓ ◆ x D p = ds = ds if L = path length along ideal trajectory ⇢ ⇢ p between 2 points, then The relative change in path length, per D(s) L ds p relative change in momentum, is called = ⇢(s) the momentum compaction factor, L R ds · p $p = < D/! > along the ideal path

Summer 2017 MJS R EBSS 2017 22 The Slip Factor

L Momentum Compaction Factor: t = v dL/L ↵ p ⌘ dp/p dt dL dv ✓ ◆ = 1 ⌘ ↵p t L v ⌘ 2[D(s)/⇢(s)]ds ↵ = dv 1 dp p ds R = 2 v p = D/⇢R D = dispersion h i dt 1 dp = ↵p dL/L t 2 p ↵ ✓ ◆ p ⌘ dp/p dt dp ✓ ◆ = ⌘ 1 The Slip Factor: ⌘ ↵ t p ⌘ p 2

Summer 2017 MJS EBSS 2017 23 A Simple Example… d

R/R0 = p/p0

p p0 p p0 v 1 p = v 2 p R R0 R R0

⌧ =2⇡R/v ⌧ =(2⇡R +2d)/v

L R p L 2⇡(R R0) 1 p = = = = L0 R0 p0 L0 2⇡R0 +2d 1+d/⇡R0 p0

⌧ 1 p ⌧ 1 1 p =(1 ) = ⌧ 2 p ⌧ 1+d/⇡R 2 p 0 0 0 0 ✓ 0 0 ◆ 0

Summer 2017 MJS EBSS 2017 24 Implications of the Slip Factor

§Suppose no bending in the line (e.g., linac), or, perhaps have bending yet γ2 < 1/αp • then, the slip factor is negative, and particles of higher momentum take less time to traverse the same distance as the ideal particle 1 D 1 ⌘ = ↵ = p 2 ⇢ 2 ⌧

§ If the energy of the particles is high enough in the presence of bending, then can have γ2 > 1/αp • in this case, the slip factor is positive — the changes in path length outweigh the changes in speed when determining the time of flight difference • here, a higher-momentum particle will actually take longer to traverse the same distance as the ideal particle, even though it’s moving faster

Summer 2017 MJS EBSS 2017 25 Linear Motion Very Near the Ideal Particle §Particles moving along the ideal trajectory move toward or away from the ideal particle according to their speed (momentum/energy) and path length differences

∆t = arrival time relative to the ideal arrival time (∆t = 0) z z = c t z0

L ⌧ = L/v = L/(c) p z = z ⌘L 0 p L p p 1 E 1 1 W t = t0 + ⌘ Note : = = c p p 2 E 2 W

Summer 2017 MJS EBSS 2017 26 Linear Motion Very Near the Ideal Particle [2]

§Imagine a particle on the ideal trajectory and that has the ideal energy, Ws. A second particle on the ideal trajectory, but with a different energy, W, may be ahead of or lagging behind the ideal particle. §We will use radio frequency (RF) cavities to provide an accelerating voltage to the particles as they pass by. §The ideal particle will arrive at the cavity at the “ideal” time or, equivalently, at an ideal phase, %s, to receive an appropriate increase in its energy (which might be an increase of “0”). §We will keep track of the “difference” in energy between our test particle and the ideal particle: Ws = “ideal” energy W W W ⌘ s

Summer 2017 MJS EBSS 2017 27 Acceleration using AC Fields

§Pass through a gap with an oscillating field…

§ W = qEd= qV W/A [eV/u] = (Q/A) eV

§ But here, V is an “average” or “effective” potential; depends upon the frequency of the field in the gap, the incoming speed of the particle (due to the field varying with time), and the phase of the oscillation relative to the particle arrival time:

W [per nucleon] = (Q/A) T () eV0 cos()

Summer 2017 MJS EBSS 2017 28 Accelerating Gap

§Create electric field / potential across two “plates”

§Vary the field with frequency, f

D. Alt, et al., MSU

Summer 2017 MJS EBSS 2017 29 Transit Time Factor §

1 g/2 T ()= E(0,s) cos(ks)ds V0 g/2 Z

g/2 2⇡ V0 = E(0,s)ds k g/2 ⌘ Z

§For v = c, and for a gap = &/2, the TTF will be • Once get up to higher velocities 1 /4 2 (v ~ c), then can consider cos(2⇡z/)dz = multiple-cell cavities since the /2 /4 ⇡ Z velocity is no longer changing.

Summer 2017 MJS EBSS 2017 30 What are low- superconducting resonators? Resonant Structures low- cavities: Just cavities that accelerate efficiently particles with <1; low- cavities are often further subdivided in low-, medium-, high-

=1 SC resonators: .elliptical3 shapes

Courtesy A. Facco

<1 resonators, from very low (~0.03) to intermediate (~0.5): many different shapes and sizes

Summer 2017 MJS EBSS 2017 A. Facco ( INFN Introduction to Superconducting Low-beta Resonators MSU 23/11/201031 Normal vs.Normal Superconducting vs. Superconducting Cavities cavities

DTL tank - Fermilab Normal conducting Cu cavity @ 300K -3 Rs ~ 10 Q~104

Superconducting Nb Cavity @ 4.2K -8 Rs ~ 10 LNL PIAVE 80 MHz, =0.047 QWR Q~109

Superconductivity allows • great reduction of rf power consumption even considering cryogenics (1W at 4.2K ~ 300W at 300K) • the use of short cavities with wide velocity acceptance

A. Facco –FRIB and INFN SRF Low-beta Accelerating Cavities for FRIB MSU 4/10/2011

Summer 2017 MJS EBSS 2017 32 Quarter-wave StructuresQuarter-wave stuctures: small g/, small size

§Imagine a coaxial L ~/4 waveguide set to transmit Z0 =V0/I0 characteristic impedance Ι an EM wave of frequency I0 V V0 C L Tg(L/c) ~ 1/(CLZ0) f, and wavelength & = c/f U ~ V0/(8 Z0) stored energy §Make the waveguide of I 1 V length L = &/4, and “close” 0 0 one end Vz() 0.5 V ~ V0sin(z/c)sin(t) Iz() • create a “standing I~Icos(z/c)cos(t) wave” within the 0 0 structure 00.20.40.60.8L § At the end where the E z field is strong, allow the beam to pass through B E two gaps • separate gaps by distance #&/2

A. Facco ( INFN Introduction to Superconducting Low-beta Resonators MSU 23/11/2010 Summer 2017 MJS EBSS 2017 33 Half-waveHalf-wave Structures structures -- More / more Symmetry symmetry

Vz() 0 Iz()

1 00.511.52L ~/2

ΙΙV0 CL

U ~ 2V 2/(8 Z ) 0 0 !A half-wave resonator is equivalent to 2 QWRs facing each other and connected ! The same accelerating voltage is Summer 2017PHWR MJS~2 PQWR EBSS 2017obtained with about 2 times larger power34

A. Facco ( INFN Introduction to Superconducting Low-beta Resonators MSU 23/11/2010 Transit Time Factor for 2-gap '-mode Cavity T() for 2 gap ( mode)

Ez Ez (constant Ez approximation) d

πg sin βλ d T ()β ≅ sin π g πg βλ lint βλ 1 L 0.5 T() 1° term: 1-gap effect g</2 0 Optimum β 2° term: 2 gap effect d~/2 1°+ 2° term TTF curve 0.5 (For more than 2 equal gaps in 0.2 0.4 0.6 0.8 mode, the formulas change only in the 2° term)

A. Facco ( INFN Introduction to Superconducting Low-beta Resonators MSU 23/11/2010

Summer 2017 MJS EBSS 2017 35 Use in Heavy ion Accelerators

Would like ability to accelerate various isotopes, i.e., a variety of energies and Q//A ratios

FRIB will use a variety of cavity frequencies, with each cavity voltage and phase being independently variable

Summer 2017 MJS EBSS 2017 36 Multi-cell Cavities

§Here: “ILC” (International Linear Collider) 9-cell style cavity as v —> c, can use multiple cells in succession cells space by RF half-wavelength

1 /4 2 for v = c, the TTF will be… cos(2⇡z/)dz = /2 /4 ⇡ Z have achieved > 35 MV/m average accelerating gradient with superconducting cavities (Note: even larger gradients achieved with non-SC, but very power intensive)

Summer 2017 MJS EBSS 2017 37 Linear Motion Very Near the Ideal Particle

§If a group of particles passes through a cavity such that the ideal (synchronous) particle receives no net energy gain, can give particles that are ahead/behind a decrease/increase in energy

Ws = “ideal” energy

1.0 VRF W W Ws

0.5 ⌘ 0.0 sin(x) -0.5 -1.0 0 10 20=2⇡fRF t 30 40 50 W = W + qV (sin sin ) x 0 s W = W + qV [sin( + ) sin ] W + qV cos 0 s s ⇡ 0 s W W + qV cos = W + qV cos (2⇡f )t ⇡ 0 s 0 s RF 0 §Can use matrix techniques to propagate the longitudinal motion

Summer 2017 MJS EBSS 2017 38 Linear Motion through Cavities and Drifts

§Keep track of time differences and energy differences…

L 1 1 t 1 ⌘ 3 2 t drift: = c mc W 01 W ✓ ◆ ✓ ◆✓ ◆0 t 10t through cavity: = longitudinal focusing W (2⇡f )qV cos 1 W ✓ ◆ ✓ RF s ◆✓ ◆0 dL/L ↵ p ⌘ dp/p remember — ✓ ◆ 1 [D(s)/⇢(s)]ds ⌘ ↵ ↵ = ⌘ p 2 p ds R = D/⇢R Summer 2017 MJS EBSS 2017 h i 39 Bunchers, Re-bunchers, Debunchers

§If start with continuous stream of particles (DC current, with no strong “AC” component), can create bunches (AC beam) using a single cavity (buncher)

§If have bunched beam that is allowed to travel a certain distance, the particles within the bunch will begin to spread out due to the inherent spread in momentum • re-buncher: mitigate this effect • debuncher: enhance this effect

Summer 2017 MJS EBSS 2017 40 Beam Buncher

5*dW_rms = 24.824 eV/u Q/A = 0.25 W_width 469.8 eV/u 1.0

incoming DC beam 4 after the cavity 0.5 2 0 0.0 dW/W [%] dW/W [%] 2 − 0.5 − 4 − 1.0 − −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 dt [ns] dt [ns]

V_1 = 900 V, at 16.1 MHz 5*dt_rms = 62.221 ns V_2 = 0 V, at 32.2 MHz L_focus = 2.2 m V_3 = 0 V, at 48.3 MHz 500

downstream of cavity 4 resulting time profile 400 2 300 0 Eff = 58.6 % 200 dW/W [%] Frequency 2 − 100 4 − 0

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 dt [ns] dt [ns]

Summer 2017 MJS EBSS 2017 41 Multi-harmonic Buncher §Use 2, or 3 (or 4?) harmonics of the fundamental frequency to smooth out the sine wave into a more linear waveform

V(t) = V1 sin(2πft) + V2 sin(4πft) + V3 sin(6πft) + V4 sin(8πft) + …

ReA pre-buncher; Alt, et al. (MSU)

Summer 2017 MJS EBSS 2017 42 Adiabatic Capture in a Storage Ring

Summer 2017 MJS EBSS 2017 43 Repetitive Systems of Acceleration • We will assume that particles are propagating through a system of accelerating cavities. Each cavity has oscillating fields with frequency fRF, and maximum “applied” voltage V (i.e., this takes into account TTF’s, etc.). The ideal particle would arrive at the cavity at phase %s.

• We will choose %s to be relative to the “positive zero-crossing” of the RF wave, such that the ideal particle acquires an energy gain of

E = W = qV sin s = QeV sin s

» this definition used for synchrotrons; linacs more often define %s relative to the “crest” of the RF wave • apologies for this possible further confusion…

Summer 2017 MJS EBSS 2017 44 Acceleration of Ideal Particle

§Wish to accelerate the ideal particle. As the particle exits the (n+1)-th RF cavity/station we would have

(n+1) (n) Es = Es + QeV sin s

§If we are considering a synchrotron, we can consider the above as the total energy gain on the (n+1)-th revolution. The ideal energy gain per second would be: dEs/dt = f0QeV sin s §Next, look at (longitudinal) motion of particles near the ideal particle: § § = phase w.r.t. RF system § E E E = energy difference from the ideal ⌘ s

Summer 2017 MJS EBSS 2017 45 Acceleration § Assume that our accelerating system of cavities is set up so that the ideal particle always arrives at the next cavity when the accelerating voltage V is at the same phase (called the “synchronous phase”) L

2⇡h⌘ En E n+1 = n + En n+1 2E n n+1 E = E + QeV (sin sin ) n+1 n n+1 s Notes: (difference equations) h = L/, = c/frf

If L is circumference of a synchrotron then: h = frf /f0 s where f0 is the revolution frequency, 1.0 In this case, h is called the “harmonic number” 0.5 0.0 sin(x) 0.5 2 − 1.0 E = mc + W ; E W − , x

Summer 2017 MJS EBSS 2017 46 Applying the Difference Equations

while (i < Nturns+1) { phi = phi + k*dW dW = dW + QonA*eV*(sin(phi)-sin(phis)) points(phi*360/2/pi, dW, pch=21,col="red") i = i + 1 }

Let’s run a code…

Summer 2017 MJS EBSS 2017 47 Summer 2017 MJS EBSS 2017 48 Acceptance and Emittance §Stable region often called an RF “bucket” •“contains” the particles §Maximum vertical extent is the maximum spread in energy that can be accelerated through the system

Summer 2017 MJS EBSS 2017 49 Acceptance and Emittance §Stable region often called an RF “bucket” •“contains” the particles §Maximum vertical extent is the maximum spread in energy that can be E accelerated through ∆ the system §Desire the beam particles to occupy much smaller area in the phase space

∆t area: “eV-sec” Note: E, t canonical

Summer 2017 MJS EBSS 2017 50 Golf Clubs vs.Golf Fish Clubs vs. Fish

! Our analysis “assumes” slowly changing variables (including the energy gain!). Quite reasonable in many Alvarez-style linacs and in synchrotrons; not as reasonable an assumption in our case ! In linacs, fractional energy change can be large, and so this will distort the phase space

! Plots from Wangler’s book: Here, assume that energy is Here, a more “constant” or rapid acceleration varying very is included slowly (linac) (synchrotron)

Summer 2017 MJS EBSS 2017 51 M. Syphers 10 Oct 2011 3 Motion Near the Ideal Particle

Linearize the motion, and write in matrix form… 2⇡h⌘ = + E n+1 n 2E n E = E + QeV (sin sin ) n+1 n n+1 s = E + QeV (sin cos +sin cos ) sin ) n s n+1 n+1 s s = En + QeV cos s n+1 2⇡h⌘ = E + QeV cos + E n s n 2E n  Thus, 2⇡h⌘ = + E n+1 n 2E n 2⇡h⌘ E = QeV cos + 1+ QeV cos E n+1 s n 2E s n ✓ ◆

Summer 2017 MJS EBSS 2017 52 or, 2⇡h⌘ 1 2E = E QeV cos 1+ 2⇡h⌘ QeV cos E ✓ ◆n+1 s 2E s ! ✓ ◆n ⇣ ⌘ 101 2⇡h⌘ = 2E QeV cos s 1 01 E ✓ ◆✓ ◆✓ ◆n

. M = Mc Md “thin” cavity drift (acts as longitudinal focusing element)

Note: for ( < 0, Md is a “backwards” drift; i.e., "% decreases for "E>0 (when no bending) ( = -1/)2 in straight region (linac)

Summer 2017 MJS EBSS 2017 53 Outlook for the Field

Summer 2017 MJS EBSS 2017 54 Advancing Critical Currents in Superconductors

University of Wisconsin-Madison Applied Superconductivity Center

note: “engineering” current densities will be less than these “critical” current densities

Summer 2017 MJS EBSS 2017 55 carried out at the LBNL test facility, are shown in 16 Figure 12 indicating low and incomplete training Short sample limit 15 curves. Most quench origins were located at the end of Accelerator Magnets the straight sectionquench training: just prior to the start up of the bend. 14 A subsequent autopsy at that location showed an 13

unintended step in the upper block (Fig. 13) created by 12 the cable hard-way bend. In HD3 coils, under construction, the radius of the bend was increased to 11 magnetic pressure: 2 ease the bend and a supporting “membrane” was added field (T) Bore 10 between layers. Other test results including strain 9 HD2a B gauges measurements, training performance, quench HD2b locations, and ramp-rate studies are reported in [5]. 8 HD2c Other improvements now include curing of coils (using 7 HD2d P = HD2e a binder) to better position them prior to reaction. By 6 reducing the reaction temperature of HD3 coils, a more 0 5 10 15 20 25 30 35 40 45 50 2µ Training quench # 0 conservative approach was taken by a corresponding 2 reduction of the current density from 3300 A/mm Figure 12: Bore field (T) as a function of training 2 (12 T, 4.2 K) in HD2 to 3000 A/mm in HD3. The quenches. The short sample limit of 15.6 T bore field impact of all such changes reduced the short-sample corresponds to a coil peak field of 16.5 T. 2 bore field from 15.6 T in HD2 to 14.9 T in HD3. (4 T) 2 7 =6.4MN/m = 64 atm 8⇡10 (13 T)2 7 = 670 atm 8⇡10

Figure 13: Cross-section cuts of HD2 coil #1 close to the beginning of the hard-way.

Analysis Figure 10: Magnet HD2 LBNL has been developing 3D finite element models to predict the behaviour2 of high field Nb3Sn (16superconducting T) magnets [6]. The models track the coil response during assembly,= cool 1000-down and excitation atm with 2 particular interest7 on displacements when frictional forces (8 T) 8arise.⇡ As10 Lorentz forces can be cycled and irreversible = 250 atm displacements can be computed and compared with strain 7 gauge measurements. Analysis on the release of local 8⇡10 frictional energy 2during magnet excitation results in a (20temperature T) increase that can be calculated. Magnet quenching and training is then correlated to that level [7]. Figures 14-15 show7 the= results 1600 of the analysis atm using the 8programs⇡10 TOSCA and ANSYS.

Figure 11: Computed field magnitude of HD2

Summer 2017 MJS EBSS 2017 56 The essential role of the superconductors in the search for higher energy

Advanced Magnet Design Texas A&M, LBNL, BNL, CERN …Proceedings of MT-23

2 Jeng 400 A/mm

Nb-Ti up to 8 T

§“stress management” Nb3Sn up to 13 T •“block” coils HTS up to 20 T •end designs are critical HTS: more than 1500 tons procurement by 2030 Concept and models now

§Canted Cosine Theta Design A. Ballarino 34 LTSW2013, 4/11/2013

nested arrangement of canted coils can possibly reach fields up to 16-20 T

LBNL

Summer 2017 MJS EBSS 2017 57 Accelerator Cavities

§ SRF (efficient, 30-50 MV/m) § vs. Cu (100-150 MV/m)

§ development of production techniques • surface treatments, doping of Nb to reduce Q-slope, etc.

§ Nb-coated materials — lower cost SRF? Grassellino, et al.

Summer 2017 MJS EBSS 2017 58 Some Projects — Current and Envisioned § FRIB (MSU) 200 MeV/u, 400 kW Facility for Rare Isotope Beams § LCLS-II (SLAC) 4 GeV Linac Coherent Light Source upgrade § FAIR (GSI) 34 GeV/u, 100 kW § ESS (Lund) 5 MW European Spallation Source (2 GeV p linac) § PIP-II (FNAL) 800 MeV p, MW-scale linac § ILC (?? Japan ??) 500 GeV electron-positron collider § CLIC (?? CERN?? ) 3 TeV electron-positron collider § e- — ion collider (?? BNL ??) 5-20 GeV e- / 50-250 GeV ions § FCC (?? CERN ??) 100 km Future Circular Collider 100 TeV pp § China: 50-80 km ring(s), 240 GeV ee collider; 70-100 TeV pp § DAEdALUS cyclotrons, 800 MeV § NuSTORM / NuFactory / Muon Collider — neutrino, muon sources § …

Summer 2017 MJS EBSS 2017 59 Future Applications of Accelerators

§Energy — ADS §Homeland Security §Defense §New Medical Techniques §Isotope Production §New industrial applications?

Summer 2017 MJS EBSS 2017 60 Extrapolating to the Future: Livingston Plot §In 1954, M. Stanley Livingston produced a curve in his book High Energy Accelerators, indicating exponential growth in particle beam energies over “past” ~25 years;

•the 33 “Bev” (GeV) AGS at Brookhaven and 28 GeV PS at CERN were underway, and kept up the trend

§The advent of Strong Focusing (A-G focusing) was key to keeping this trend going...

Summer 2017 MJS EBSS 2017 61 The Past 40 Years

Summer 2017 MJS EBSS 2017 62 Some limiting factors ...

§superconducting technology -- accel. cavities this time, not magnets §high accelerating gradient (>35 MeV/m) §Synchrotron Radiation •effects obvious in e+e-; hence, the L in ILC •real estate vs. electric field strength §stored energy an issue in LHC; beam power issue in linac §energy deposition in targets, Interaction Points; backgrounds §small apertures --> alignment tolerances (micron scale) §requires very small beam sizes — approaching nm scale •damping rings -- S.R. put to good use •emittance exchange -- eliminate need for damping rings?

Summer 2017 MJS EBSS 2017 63 The Livingston Curve Again §In attempt to compare adopted from W. Panofsky. Beam Line (SLAC) 1997

e- & p, switch to C-of-M SSC (÷ 8) LHC view of constituents ? §seeing a new roll-off happening ILC §driven by budgets, if constrained to present technology §thus, need new technologies to make affordable...

Summer 2017 MJS EBSS 2017 64 Lawrence Berkeley Lab Laser Wakefield Acceleration Research News: From Zero to a Billion Electron Volts in 3.3 Centimeters - Highest Energies Yet From Laser Wakefield Acceleration 01/30/2007 12:26 AM

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September 25, 2006 news releases | receive our news releases by email | science@berkeley lab

From Zero to a Billion Electron Volts in 3.3 Centimeters Highest Energies Yet From Laser Wakefield Acceleration

Contact: Paul Preuss, (510) 486-6249, [email protected]

BERKELEY, CA — In a precedent-shattering demonstration of the potential of laser-wakefield acceleration, scientists at the Department of Energy's Lawrence Berkeley National Laboratory, working with colleagues at the University of Oxford, have accelerated electron beams to energies exceeding a billion electron volts (1 GeV) in a distance of just 3.3 centimeters. The researchers report their results in the October issue of Nature Physics. •30 GeV/m, compared to 30 MeV/m in present SRF By comparison, SLAC, the Stanford Linear Accelerator cavity designs Center, boosts electrons to 50 GeV over a distance of two miles (3.2 kilometers) with radiofrequency cavities whose accelerating electric fields are limited to about 20 million •... and, small momentum spread (2-5%) as well volts per meter.

The electric field of a plasma wave driven by a laser pulse can reach 100 billion volts per meter, however, which has made it possible for the Berkeley Lab group and their Oxford collaboratorsAlso, to similar achieve a 50th of(but SLAC's beam different) efforts at energy in just one-100,000th of SLAC's length.

This is only Uthe firstTexas, step, says Wim U Leemans Michigan, of Berkeley Stanford/SLAC, Lab's Accelerator and Fusion Research Division (AFRD). Billion-electron-volt, high-quality electron beams "Billion-electron elsewhere...-volt beams from laser-wakefield have been produced with laser wakefield acceleration in recent experiments by Berkeley accelerators open the way to very compact high-energy Lab's LOASIS group, in collaboration with experiments and superbright free-electron lasers." scientists from Oxford University.

Channeling a path to billion-volt beams

In the fall of 2004 the Leemans group, dubbed LOASIS (Laser Optics and Accelerator Systems Integrated Studies), was one of three groups to report reaching peak energies of 70 to 200 MeV (million electron volts) with Summer 2017 laser wakefields, accelerating bunches ofMJS tightly focused electrons with nearly uniform EBSS 2017 65 energies.

While the other groups employed large laser spot sizes and 30 TW laser pulses (TW stands for terawatts, or 1012 watts), the LOASIS "igniter-heater" approach was quite different. LOASIS drove a plasma channel http://www.lbl.gov/Science-Articles/Archive/AFRD-GeV-beams.html Page 1 of 5 Lawrence Berkeley Lab Laser Wakefield Acceleration

§Staging Demo

Summer 2017 MJS EBSS 2017 66 In the Meantime, … FCC?

§ Coming off of the successful first running of LHC, and the Nobel-Prize- winning Higgs discovery, Europe is engaging the international community in discussions/studies of a Future Circular Collider § China is also looking at large rings for its future

Parameters are a 50 TeV x 50 TeV p-p collider, with circumference ~100 km (about same size as the Texas SSC, which was a 20 TeV x 20 TeV collider)

View from France into Switzerland, showing existing LHC complex (orange) and a possible 100 TeV collider ring (yellow)

photo courtesy J. Wenninger (CERN)

Summer 2017 MJS EBSS 2017 67 This relationship can be expressed across a potential difference of one quantitatively. To examine matter at volt.) At the scale of the nucleus, en- the scale of an atom (about 10!8 cen- ergies in the million electron volt— timeter), the energies required are in or MeV—range are needed. To ex- the range of a thousand electron amine the fine structure of the basic A “Livingston plot” showing accelerator energy versus time, updated to include volts. (An electron volt is the energy constituents of matter requires en- machines that came on line during the unit customarily used by particle ergies generally exceeding a billion 1980s. The filled circles indicate new or physicists; it is the energy a parti- electron volts, or 1 GeV. Looking Below the Curve upgraded accelerators of each type. cle acquires when it is accelerated But there is another reason for using high energy. Most of the objects of interest to the elementary particle physicist today do not exist as free 1000 TeV particles in Nature; they have to be created artificially in the laboratory. §Accelerator Facilities, and the need The famous E = mc2 relationship gov- 100 TeV erns the collision energy E required for scientists to develop, build, to produce a particle of mass m. Proton Storage Rings Many of the most interesting parti- commission, operate, improve them 10 TeV (equivalent energy) cles are so heavy that collision energies of many GeV are needed to have seen an enormous growth create them. In fact, the key to under- 1 TeV standing the origins of many para- over the decades meters, including the masses of the Proton known particles, required to make Synchrotrons

y today’s theories consistent is believed

g 100 GeV r

e to reside in the attainment of colli- §While peak accelerator energies n E sion energies in the trillion electron e l c

i volt, or TeV, range. t Electron

continue to drive particle physics, r 10 GeV a Synchrotrons Our progress in attaining ever P Electron Linacs higher collision energy has indeed much work to do and applications Synchrocyclotrons Betatrons been impressive. The graph on the 1 GeV to develop at lower energies Proton Linacs left, originally produced by M. Stan- ley Livingston in 1954, shows how Sector-Focused the laboratory energy of the parti- Cyclotrons 100 MeV cle beams produced by accelerators §Many, many facilities and industrial Cyclotrons Electrostatic has increased. This plot has been up- uses are not shown here, but flood Generators dated by adding modern develop- 10 MeV ments. One of the ﬁrst things to no- the area “below the curve” tice is that the energy of man-made Rectifier accelerators has been growing ex- Generators 1 MeV ponentially in time. Starting from the 1930s, the energy has increased— roughly speaking—by about a factor of 10 every six to eight years. A 1930 1950 1970 1990 second conclusion is that this spec- Year of Commissioning tacular achievement has resulted

38 SPRING 1997

Summer 2017 MJS EBSS 2017 68 A “Final” word...

Summer 2017 MJS EBSS 2017 69 A “Final” word...

M. Stanley Livingston, 1954

Summer 2017 MJS EBSS 2017 69

THANKS!

Mike Syphers

[email protected] [email protected]

§ Further reading: • D. A. Edwards and M. J. Syphers, An Introduction to the Physics of High Energy Accelerators, John Wiley & Sons (1993) • T. Wangler, RF Linear Accelerators, John Wiley & Sons (1998) • H. Padamsee, J. Knobloch, T. Hays, RF Superconductivity for Accelerators, John Wiley & Sons (1998) • S. Y. Lee, Accelerator Physics, World Scientific (1999) • and many others… § Many Conference Proceedings — visit http://www.jacow.org