NuFACT07 Summer School Factory Front End and Cooling

1
NuFACT07 Summer School?-Factory Front Endand
Cooling

• David Neuffer
• Fermilab

2
Beam Optics, Tutorial

• Longitudinal Beam dynamics
• Equations of motion
• Discussion
• Optics transverse dynamics
• General introduction
• Betatron functions
• Focusing fields
• Quadrupoles
• Solenoids
• Li lens, Magnetic horn
• Cooling
• Sample parameters
• Cooling of ions

3
Basic Equations

• Maxwells equations
• Equation of motion

• Transport/focusing uses magnetic fields
• Dipoles, quads, solenoids,
• Horns, Li lens,
• Acceleration uses rf electric fields
• rf cavities
• Induction modules

4
Magnetic motion

• Force is perpendicular to motion
• ?, ßv/c, P remain constant
• If v, B are perpendicular, motion is circular
• m?v/q BR P/q is called the magnetic rigidity
  or B?
• 1 GeV/c particle has B?109/c 3.33 T-m
• Bending angle in magnet with field B, length L is
  ?BL/ B?

Bending radius
What is B? for a 300 MeV/c muon?
5
Homework problems

• Target
• How many 24 GeV protons per second are in a 4 MW
  beam?
• PRISM
• Homework rewrite equations of motion in F-dp
  units (ddp/p)

6
Longitudinal Beam Dynamics

• Longitudinal equations of motion (????)

In dz-dE/mc2 units
7
Longitudinal motion stuff

• Distance for oscillation
• Adiabatic ( if L gt ?osc)
• Longitudinal Cooling equations

• Rf Bucket energy width

8
Betatron Functions

• Assume focusing is linear around the orbit
• Harmonic oscillator ( k(s))
• Solution
• where

• Focussing forces
• Quads kx(s) B(s)/B?
• ky(s) - kx(s)
• Solenoids k(s) (B/2B?)2
• With x-y rotation
• Need starting amplitude and derivative to specify
  motion
• ßx(0), ßx'(0)
• In periodic structure these are set by requiring
  that ßx(s) be periodic.

At equilibrium ßx (1/k)½
Exercise show that these equations are consistent
9
At a focus (with kx0)
To minimize beam heating Labsorber lt
ß? unless kx is non-zero
10
Solenoidal Focussing

• Magnetic field
• Cylindrical symmetry

• Change to R-f notation
• Solution is rotation in f, focussing in R

11
Comparison of focussing

• Quadrupole
• Focuses x or y
• Proportional to 1/P
• Solenoid
• Focuses both x and y
• Focuses both ? and ?-
• Proportional to 1/P2
• Solenoid better for low-energy
• For B? lt Bo a (1 T-m ?)

12
Betatron function discussion

• Beam optics typically described in terms of
  betatron functions
• Use rms Emittance (beam phase space area)
• Emittance also defined as
• References are often unclear on whether
  normalized or normalized emittance is used also
• Fermilab convention ?f 6p?rms
• CERN convention ?CERN 4p?rms

13
Solenoidal lattices (with field flips)

• Magnets used
• FOFO lattice
• Used in study 2A (B2.5T)
• Produces constant focusing
• ß? ? Lcell
• Super FOFO
• Used in study 2
• Bmax ? 5.5T
• Produces small ß? between strong coils
• ß? ? 0.2m
• Single flip, double flip
• ß? ? 2 B?/B

14
Magnetic horn

• Current along horn creates magnetic field B?
  outside surface (B0 inside)
• Focussing field from Fr qvzB?
• If horn profile is parabolic, ?z?r2, integrated
  force is linear
• Focuses only one sign
• S. van der Meer

NUMI horn 205 kA peak current 3.0 Tesla peak
field 2.6 milli-sec half-sine wave pulse 1.87 sec
repetition rate
15
Li-lens cooling

• Lithium Lens provides strong-focusing and low-Z
  absorber in same device
• Liquid Li-lens may be needed for highest-field,
  high rep. rate lens
• BINP (Silvestrov) was testing prototype liquid Li
  lens for FNAL
• But FNAL support was stopped - and prototypes
  were not successful ...

ß? 0.026m (200 MeV/c, 1000 T/m) ß? 0.004m (40
MeV/c, 8000 T/m)
16
Adiabatic damping and normalized emittance

• Under acceleration
• dx dpx remain constant (normalized emittance)
• dx dx' dx dpx/p is proportional to (1/p)
• geometric emittance decreases (beam size
  decreases)
• Similarly,
• dct dE remain constant
• dz dp/p decreases as 1/p (acceptances set by
  dp/p)
• Acceptances improve as beam is accelerated
• Use higher-frequency rf, weaker transverse
  focusing

17
Cooling requirements

• Set by acceptance of downstream accelerators and
  storage ring
• Study 2 200MHz, Pµ 200MeV/c
• ex,max lt 0.015m, ey,max lt 0.015m, ez, maxlt0.15m
• xmax,, ymax lt 8.5cm (at ß?1m), zb lt 0.5m
• ex,rms lt 0.06
• Study 2A 200MHz, Pµ 200MeV/c
• ex,max lt 0.03m, ey,max lt 0.03m, ez, maxlt0.15m
• xmax,, ymax lt 12cm (at ß?1m), zb lt 0.5m
• ex,rms lt 0.12
• May change in future redesign

18
Homework Problem

• 1300MHz Superconducting rf is cheap and
  available
• Internal aperture is r3cm
• Assume
• 1300 MHz rf is used in 1m long cavities, with a
  quad magnet between cavities , 1.3 m half-cell
  (ß?,max 4m)
• How small must the emittance be to allow muon
  acceleration ?
• Require 2.5 ? aperture
• Guesstimate an acceptable injection energy/
  cooled emittance
• Redo exercise for 800MHz (r5cm)

19
Problem

• Plug in some sample values for the cooling
  equations solve for ?t equilibrium and ?t(s)
• try dE/ds 5 MV/m, p 300 MeV/c,
• ?t 0.1m, Li absorbers ?N(0)0.01

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Solution

• Lithium density is 5/84.8 of solid

21
Low-energy cooling of ions

• Ionization cooling of protons/ ions has been
  unattractive because nuclear reaction rate is
  competitive with energy-loss cooling rate
• And other cooling methods are available
• But can have some value if the goal is beam
  storage to obtain nuclear reactions
• Absorber is also nuclear interaction medium
• Y. Mori neutron beam source
• NIM paper
• C. Rubbia, Ferrari, Kadi, Vlachoudis source of
  ions for ß-beams

22
Cooling/Heating equations

• Cooling equations are same as used for muons
• mass a mp, charge z e
• Some formulae may be inaccurate for small ßv/c
• Add heating through nuclear interactions
• Ionization/recombination should be included
• For small ß, longitudinal dE/dx heating is large
• At ß 0.1, gL -1.64, ?g 0.36
• Coupling only with x cannot obtain damping in
  both x and z

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Example

• ERIT-P-storage ring to obtain directed neutron
  beam (Mori-Okabe)
• 10 MeV protons
• 10Be target for neutrons
• ? v/c 0.145
• Large dE heating

24
RMS Cooling equations
For small ß
Better for larger mass?
25
Problem set

• Use above discussion and equations to set up
  cooling constraints and possible parameters for
  Li6 cooling by He3.
• Are the phase-space mixed heating terms greater
  than the direct terms ? (see Wang and Kim)
• Track cooling with an appropriate tracking
  code compare with analytic numbers
• Suitable for publication/Ph. D. thesis

26
ß-beams example 6Li 3He ? 8B n

• Beam 25MeV 6Li
• PLi 529.9 MeV/c B? 0.59 T-m v/c0.09415
  Jz,0-1.6
• Absorber3He
• Z2, A3, I31eV, z3, a6
• dE/ds 110.6 MeV/cm,
• (?He-3 0.09375 gm/cm3)if Liquid,
• dE/ds 1180 MeV/gm/cm2, LR 70.9 gm/cm2
• If gx,y,z 0.13 (Sg 0.4), ß- 0.3m at
  absorber
• Must mix both x and y with z
• eN,eq 0.000046 m-rad,
• sx,rms 2 cm at ß- 1m
• sE,eq is 0.4 MeV
• Promising but many problems

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References

• Beam transport codes (linear optics fitting)
• TRANSPORT, MAD, DIMAD, OPTIM, COSY
• Not always good for unusual optics
• Do not include absorber effects
• Tracking codes
• ICOOL and DP4GEANT
• Accelerator physics texts S. Y. Lee, Edwards and
  Syphers,etc.
• Handbook of Accelerator Physics and Engineering
  (A. Chao and M. Tigner, eds.)