# Presentation: 669713

## ISS Muon Bunch Structure 

*Convenors: S Berg, BNL, and G H Rees, RAL*

## ISS 201 MHz Bunch Structure Options (Separate comparison later for lower frequencies)

*Single train of 80 **μ**+** and 80 **μˉ** bunches/cycle?*
*Or, n trains of 80 **μ**+** and 80 **μˉ** bunches/cycle?*
*Factor of 50/3 in **μ**±** currents proposed to date:*
*n = 1 at F = 15 Hz,        I/nF = 50/750   (USA)*
*n = 5 at F = 50 Hz,        I/nF =   3/750   (RAL)*
*  **What are the criteria for choosing n and F?*

## Factors involved in choice of n and F

*                                                    **Preferred values*
*Target thermal shock effect:     high F and energy*
*Design of the proton driver:      n =3, 4 or 5 ; F = 50 Hz*
*Beam loading in **μ**±** stages:       high n, F and radius *
*Switch-on power for **μ**±** RF:       low F (P scales with F)*
*Design of **μ**+**, **μˉ** **decay rings:    n (inj) <6 ; high F (RF) *

## Target Effects (SB & RB)

*(SB)** π**± ** and **μ**±** yields: 10 GeV a good proton energy*
*                                     **(except for a carbon target?)*
*(RB) Solid Targets: Thermal shocks reduced at higher         *
*                               **F, & by 50 **μ**s delays of n bunches.*
*        **Liquid Target: The total duration of the proton    *
*                               **pulse/cycle must be < 60 **μ**s.*
*                      **.*

## Proton Drivers for the Two Cases 

*(n, F(Hz), T(GeV))   =    (1, 15, 26)          (5, 50,10)*
*Protons per bunch   =     6.4 x10**13                  **10**13 *
*Booster bunch L**b       **=     6.4 x L**b                       **L**b *
*Bunch A (eV sec)    =     6.4 x A                A (= 0.66)*
*Booster harmonic    =     1                         6*
*Driver harmonics    =   ~ 6, 36, 216          36, 216*
*Final bunch **Δ**p/p    =   **± 2.0%                ± 0.8%*
*Bunch **Δ**t (ns, rms) =   < 3 ?                      2     1*
*Δ**t  compression      =      problematic       adiabatic                               *

## Proton Driver Longitudinal Bunch Area

*The bunch area to be compressed (in eV sec) is:*
*  **A* =* (8R**α**/(ch)) ((2 V(I-********sc**)E**o******) / (h**********))**1⁄2** *
*Choose low linac energy & booster radius for A < 0.7.*
*   *
*Choose 200 MeV linac & 63.777 m, booster radius.*
*Choose n =  5, h = 6, and 10**13 **protons per bunch. *
*(These allow room for the RF and ease extraction).*
*Choose h = 36 & 216, and 2 x radius for an NFFAG. *

## Proton Driver Longitudinal Space Charge

*******sc** **is the ratio of the longitudinal space charge*
*      **forces to the focusing forces of RF system.*
*******sc** **= 1 corresponds to an RF bucket collapse.*
*      **= 0.4 gives onset of a microwave instability.*
*******sc** **= 0.21 for V =   93 kV at 0.20 GeV, and*
*      **= 0.39 for V = 476 kV at 3 GeV in booster.*
*******sc** **= 0.11 for V = 469 kV at 3 GeV in NFFAG and*
*          **cancels the inductive wall fields at 10 GeV.*
*  **V  =1.0, 2.5 MV (h=36, 216) at 10 GeV (1 ns rms). *
*** ***

## Proton Driver Transverse Space Charge 

*   **Assume a 2-D elliptic beam density distribution.*
*   **Δ**Q**v    **=  - N r**p **G F / [ **π** **ε**v **(1 + a/b) **β** **γ**2 **B**f  **]*
*      **ε**v     **=  normalised, 2**σ** vertical beam emittance*
*      **G   =   1.2  and F = image force enhancement.*
*      **ε**v    **=   122 (**π**) mm mrad (2**σ**, normalised)*
*  **Δ**Q**v    **= - 0.35 in booster after Hˉ injection   *
*   **Δ**Q**v    **= - 0.25 in driver at 3 GeV injection*
*  **Δ**Q**v    **= - 0,14 at 10 GeV after compression.*

## 10 GeV, 50 Hz, 4 MW Proton Drivers

*180 MeV Hˉ **linac** + 50 Hz boosters + 2, 25 Hz RCS*
*180 MeV Hˉ **linac** + 50 Hz booster + 50 Hz NFFAG(I)*
*A Hˉ linac feeding a chain of 50 Hz FFAGs in series *
*For 1, a slower RCS needs more difficult boosters.*
*For 2, electron models are needed for both options *
*For 3, injection of Hˉ into the first FFAG is difficult.*
*Typical number of bunches are n = 4, 5, .......or 1*
*8 GeV, 50 Hz, H**ˉ **linac + accumulator + compressor?  *

## NFFAGI Proton Driver

*Alternative is a larger,*
*h=6,n=5 RCS & NFFAG*

## Proton and Muon, 50 Hz Bunch Trains

*                                                                               **O**                                                                                                                                              *
*Proton booster (n=5, h=6)                                  O        O                                                                     *
*                                                                              **O    O**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 *
*Proton driver (n=5, h=36)                                              O                                 *
*                                                                                             **O                                                                                                                                                                                                                                                             *
*                                                                             **O   O   O**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    *
*Proton bunches at target                O          O          O          O    **T **   O*
*Pion bunches after target               ****O          O          O          O          O***
*Muon, 400 ns bunch trains**                                                           *
*                                                                                                                          *
*                                                                                                                                                                                                                                                               **                               *
*(**n-1)T< 60 **μ**s (liquid target)                                                                     T                                        *
*T > 60 **μ**s (for solid targets)*                                 *μ**+          **μˉ*
*20 GeV **μ**+** & **μˉ** accelerator*
*20/50 GeV **μ**+** decay ring                      600                    600                   600**                                                                                                                                  *
*C>1500 m circumference               **400 ns bunch trains & 600(+) ns gaps*
*20/50 GeV **μˉ** decay ring**                                         **600                    600**                                                                                         *

## Box-car Stacking for Decay Rings

*Driver has protons, while muons are to be stacked.*
*So, a revised method of box-car stacking is needed.*
*Sequential delays for proton bunches ~ 30 - 70 **μ**s,*
*and an unchanging delay through the muon stages.*
*Times insufficient to adjust 201.25 MHz RF phases.*
*Make 201.25 MHz a harmonic of driver at 10 GeV.*

## n = 5, Muon Bunch Pattern in Decay Rings

- .

*2 of 5  interleaved 80 **μˉ** bunch*
*trains of the adjacent 2nd ring** *

*80 full and 127 (or 130) empty RF buckets*

*>100 ns intervals*

*80 **μ**+*

*127(130)*

*80 **μ**+*

*80 **μ**+*

*Solid/liquid148(136)*

*127(130)*

*80 **μ**+*

*127(130)*

*127(130)*

*80 **μ**+*

## Ring RF Harmonic Numbers

*     **Rings             Beta       Circ (m)         **h**   **RF (MHz)**  N**b** /Ring*
***50 GeV μ Decay        0.9999977      1573.0691****      **1056   **     **201.250 ****         5x80***
***20 GeV μ Decay        0.9999861      1573.0509****      **1056  **      **201.250****          5x80***
***20 GeV μ******± ******Acc          0.9999861       1135.0991****        **762**        **201.250 ****       10x80***
***?  GeV μ******±****** Acc                                                                          ****201.250**** ******       10x80***
***?  GeV μ******±****** Acc                                                                          ****201.250 ****       10x80***
***3-10 GeV P Driver    0.9963143        801.44744****        **36**     ****13.079-13.417          5****               *
*                                                                                  **216**           ****80.500                5***
*                                                                                  **540** **       **201.250****                5***
***0.18-3 GeV Booster  0.9712057        400.72372****         **6 **    ****2.5413-4.3595******          ******5***

## Box-car Transfer of μ+ & μˉ to Decay Rings

*The 20 GeV decay rings, 20 GeV **μ**±** acc and P driver, of periods T****d**** , T****a**** ,T****p**** ,*
*all have a harmonic at 201.25 MHz. **The integers p (= 1,2,3 ,4), n and m *
*are chosen so the proton bunch delays are a good approximation to: *
*                                                                                                               *
*                          **(n ± p/5) T****d ****  ≈  (m ± 1/12) T****p***
*T****d**** , T****a**** , T****p**** =  5.2472044, 3.7863345, 2.6832296 μs,  **(T****d**** /T****p****) =1.9555554*
*Target**   m     n      p**       **(m ± 1/12)       (n ± p/5) (T****d**** /T****p****)    |Difference**|*
*solid     23   12    -1     23 + 0.083333        23.075553         **0.007780 **                    *
*         *
*liquid     5     3     -2       5 + 0.083333**          **5.084444**          0.001111** *
*For solid target:     (m + 1/12) Tp  =  n Td  -  207 Tb   (RF period Tb )*
* *
*For liquid target:    (m + 1/12) Tp  =  n Td  -  423 Tb** *

## Summary of Proton Driver for 201.25 MHz Muon Stages (n = 4 considered later)

*Compression harder if n < 5, F < 50 Hz or T < 10 GeV.*
*The muon decay rings limit n to a maximum at n = 5.*
*Limit F to 50 Hz because of muon RF switch-on costs. *
*It does not appear necessary to increase T > 10 GeV.*
*Final bunch structure depends on accel.& target sites.  *

## 201.25 MHz Muon Stages

*1. Initial Bunch Rotation Stage (Neuffer / Iwashita)*
*    **Division into 80 bunches is needed to reduce the*
*    **longitudinal bunch areas and later beam losses.*
*             *
*2. Transverse Cooling Stage (45    30    ? (**mm rad))*
*    **Helps to reduce losses during muon acceleration.         *
*    **Lowers apertures in **μ**±** rings & transfer lines (1/√**ε**).*
*    **Lowers **μ** /** **ν** **divergence ratio in decay rings (**1/√**ε**).*
*    **Eases downstream kickers (power scales as **ε**2**).*
*           *
*3. Linac + Ring Options**: 1. RLAs, Dog-bone, DRLAs. *
*    **2. Linear, Non-scaling, Near-Isochronous FFAGs. *
*    **3. Non-linear, Not-scaling, Isochronous IFFAG(I)s.**              *

## 201.25 MHz Muon Acceleration

*No allowance for emittance growth in acceleration*
*    **Beam loss collectors needed for high power levels.*
*    **Long collimators for the counter-rotating μ**± **beams.*
*    **This infers long straights or insertions for the rings*
*Beam loading power for the rapid acceleration*
*   **This scales as 1/nFR, where R is the ring radius.*
*    **Factor of ~ 50 higher for (1, 15Hz, low R) scheme.*
*    **20 GeV ring: 1000 cf 20 units, for 2 MW couplers**.*
*Injection and Extraction Fast Kicker****  ****Systems*
*    **Large systems needed for the two decay rings.*
*    **Kickers for low R, FFAGs may not be feasible.*
*    *

## Aspects of 201.25 MHz Options

*D/RLAs:**   Kicker magnet systems not necessary.*
*                **RF systems in zero dispersion straights.*
*                **Beam loss collectors in some of the arcs?*
*FFAGs:   **Long.-transv. coupling at large amplitudes.*
*               **Is there coherent trans. motion or **ε** **growth? **  *
*               **How large does the final **Δ**p/p become?*
*IFFAGIs: **Beam losses at Q**h **= 1/3 cell resonances.*
*               **New 9.5-20 GeV design avoids this feature.** *
*               **Tracking studies haven’t yet re-commenced.*

## μ+ and μˉ Decay Rings

*Separate rings are required to allow both fast injection*
*and the time separations for the n (= 5) bunch trains.*
*For a single detector, racetrack rings are preferred.*
*For 2 detectors, two may be used, in own tunnels.*
*For two distant detectors, triangular, side by side rings*
*in vertical or near vertical plane, have higher efficiency.  *
*For detectors at 7500 & 3500 km, rings need to be in*
*a near vertical plane & to have an apex angle of ~ 50**°.*

## Features of Decay Rings

*The RF containing fields have to scale as (**Δ**p/p)**2  *
* **3, 10 MV systems needed/ring for **Δ**p/p = **± 1%*
* **Reactive beam loading compensation is needed*
*16, 50 kV PFN, 5 kA pulsers & 10 **Ω** feeders/ring*
*8, shorted, 3m, 10 **Ω** delay line, push pull kickers*
*The kicker rise and fall times have to be < 600 ns *
*Collimators in short straight of the isosceles **Δ**.*
*Use of radiation hard quadrupoles is proposed*

## Effect of n = 4 in smaller Decay Rings

*Benefits are smaller depth, cheaper tunnels for decay rings.*
*Efficiency of the two racetracks is reduced from   **38 to 35%.*
*Efficiency of two, 50**° apex** rings is reduced from **48 to 43%.*
*Options (last is favoured) for changes needed to Proton Driver:*
*1. F = 62.5 Hz; RF costs up in both Proton Driver and **μ**±** **rings.*
*2. T = 10    12.5 GeV (4 -12.5 or 3-8 and 8-12.5, GeV FFAGs).*
*3. N = 1.0    1.25 10**13**/bunch; RF costs up in Driver & **μ**±** **rings.*
*    **Lower frequency, longer cavity, RF systems are required. *
*N = 1.66 10**13**/bunch for n = 3 is also feasible (bunches longer).*

##  Lower Frequency Muon RF Systems 

*Examples:   Scaling FFAG schemes (KEK),*
*                    **44/88 MHz RF systems (CERN).*
* **KEK:*
* **A low repetition rate, 3-50 GeV, Proton Synchrotron.*
* **A chain of variable low frequency, scaling FFAGS.*
* **RF systems compensate for cavity and beam power. *
* **No transverse cooling & no separate bunch division. *
* **Apertures are enhanced in scaling FFAG magnets.        *

##   Issues for Low Frequency Muon RF

*RF systems & power costs are key considerations.*
*More space & switch-on power needed for cavities?*
*Issues little changed for the Proton Driver (n = 7?).*
*Keep F at 50 Hz to limit beam loading in **μ**±** **rings.*
*How to provide transverse cooling at low frequency?*
*Possibility of NFFAGs or IFFAGs instead of FFAGs?**                                                                                                *

## Bunch Structure Issues

*Change from 1 to **≤ 5 bunch ** trains per cycle?.*
*Use 50 Hz, 4 MW, 10 GeV Proton Driver, n=5,4,3?*
*Use proton bunch delays for low **μ** beam loading**?*
*Compare low & high frequencies for muon stages.*
*5.   Delay decision on **μ**±** acceleration for further R&D?*
*6.   Use 201.25 MHz for **μ**±** rotation, cooling & acceln?*
*7.   Create trains of 80 **μ**+** & **μˉ** bunches while rotating.*
*Accelerate the bunch trains singly in the **μ**±** rings.*
*Provide transverse cooling to give **ε** ≤ 30 mm rad?** *
*10. Two rings (racetracks) needed for single detector.*
*11. Use two vert. **Δ** rings for best **ξ** for** two detectors. *