*DRAFT, May 14, 2004*

*P.N. Ostroumov*

*ANL, Physics Division*

### Lattice design strategy for the multi-GeV SC linac 

1.  The following is based on the developments of two options for the
    cost estimate: a) RT front end and b) SC front end.

2.  Define output energy of the RFQ. This energy depends from the fast
    chopper specifications. SNS traveling wave chopper has rise- fall
    time \~10 nsec. The 2.5 kV voltage switcher should run at \~1 MHz.
    It would be technically difficult to develop similar chopper to use
    at energies higher than 2.5 MeV. If the chopper requirements for
    FNAL driver are more relaxed, the RFQ energy 5 MeV is preferable,
    especially for SC front end. Higher RFQ energy is also good to
    suppress possible source of beam halo formation due to the space
    charge in the MEBT (Medium Energy Beam Transport). MEBT has
    irregular focusing structure (long drift spaces) to accommodate
    chopper.

3.  Select frequency of the front end.

    a.  For RT front end it should be 1300:3=433 MHz

    b.  For SC option it can be either 433 MHz or 325 MHz. The option
        325 MHz will have less number of resonators and couplers, single
        spoke can be designed directly from 3 MeV
        (beta_geometrical=0.15). The 325 MHz rf system will be more
        expensive due to the larger waveguides, I guess. Klystrons are
        available from Toshiba, they have been developed for Japanese
        Hadron Facility (JHF).

4.  Select transition energy between RT front end and TESLA SC sections.
    This is not trivial. During the design of JHF linac we have done
    beam dynamics studies (see, for example,
    <http://accelconf.web.cern.ch/accelconf/l98/PAPERS/TU4106.PDF> ) of
    the 1:3 frequency transition as an option for the JHF 50 mA H-minus
    linac. The transition energy was 50 MeV and we have seen particle
    losses even with low number of simulated particles. The design of
    the JHF linac has been improved and now they are building linac with
    the transition energy 190 MeV. This energy is quite reasonable for
    1:3 frequency jump. The RT linac up to \~190 MeV can include the
    SNS-style DTL up to \~80 MeV and on RT section based on so-called
    Separated DTL (SDTL). The SDTL is similar to individual SC
    resonators. SDTLs can be coupled through the rf bridges as proposed
    by CERN SPL group, in this case it is called "CDTL"- coupled DTL. By
    the way, the SDTL/CDTL will consume at least 15 MW rf power for
    copper heating in the energy range from 80 MeV to 180 MeV, the
    length will be \~60 m.

5.  From my point of view everything above 5 MeV should be definitely
    based on the spoke resonators. At 325 MHz the input energy can be
    \~3 MeV. Select transition energy between SC front end and SC TESLA
    sections. This is simple. If SC front end operates at 433 MHz the
    transition can be anywhere above 180 MeV. For 325 MHz the transition
    is \~400 MeV. The triple-spoke definitely will be tested this year.
    It should provide operational (which means conservative) 25 MV/m
    surface field in cw mode for the RIA. The design of the TSL can be
    improved for the pulsed operation: lower peak magnetic field can be
    provided by geometry optimization of the TSR at the cost of lower
    shunt impedance. Operational surface field \~30-32 MV/m seems
    visible. If 433 MHz SC front end is selected, the design of the
    spoke resonator directly from 3 MeV is not trivial. The energy range
    from 3 MeV to \~15 MeV can be covered either with one drift tube
    resonator (like in the SNS) or with several separated RT resonators
    with SC solenoid focusing between them. High-current beam optics
    call for low-gradient acceleration below\~15 MeV due to: a) to match
    to the RFQ low accelerating field; b) to be close to the
    "equipartitioning" condition (definition is given below).

6.  After the selection of transition energies to the TESLA cavities one
    can design the squeezed TESLA (S-TESLA) cavity/cryostat. It will be
    one ~G~ for the SC front end option and 2 betas for RT front end
    option.

7.  Define input emittances --SNS numbers should be OK.

8.  If the chopper specifications are defined, start the designing of
    the MEBT. This way the design of the focusing/accelerating lattice
    can be followed from the beginning. Define beam Twiss parameters as
    in the entrance of the SNS-MEBT.

9.  Start designing of the accelerating/focusing lattice. In the
    beta=1.0 section the existing proposal (FCCCDCCC structure, C=
    cavity) seems OK. In the S-TESLA section either FCCDCC or FCCCDCCC
    should be OK. The preference must be done on the base of following
    considerations for design of high-current linacs. The beam current
    above \~5 mA should be considered as a high-current up to \~1 GeV.
    This is not "space charge dominated beam", however, bunch size
    becomes small due to the phase damping and beam can be sensitive to
    the space charge forces, especially to the transverse-longitudinal
    coupling and beam mismatch.

10. Many useful recommendations for high-current linac design have been
    developed during last 15 years (see, for example, M. Reiser's book "
    theory and design..."). Particularly important recommendations are:

    a.  ~T0~, transverse phase advance per focusing period for I=0,
        should be less than 90 deg.

    b.  Unlike in RT structures, in SC linac the length of the focusing
        period can not be changed adiabatically. However can be changed
        adiabatically. Usually the matching between the different
        sections is "current independent" if the condition is fulfilled
        (for I=0):

> $\frac{\sigma_{1}}{L_{f1}} = \frac{\sigma_{2}}{L_{f2}}$ (1)
>
> where *L~f\ ~*is the focusing period.

c.  Equipartitioning. Minimize the temperature difference between the
    longitudinal and transverse oscillations. This is especially
    important up to \~ 1 GeV.

d.  Avoid irregular focusing structure, it can produce structure
    resonances.

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11. Write a FORTRAN or C code for preliminary finding of ~T0~, ~L0~
    ,~T~, ~L~ for the given profile of accelerating gradients, length of
    focusing periods. "Equipartitioning" should be taken into account.
    At this stage realistic distributions of axial fields in all type of
    SC resonators must be known from electrodynamics codes. The goal is
    to define final focusing structure or validate the selected option.
    In the case of SC Front end the following structure is recommended:

    a.  SSR section: FsC =Focusing solenoid+cavity

    b.  DSR section: FsC

    c.  TSR section: quadrupole focusing, FCDC, maybe reasonable FCCDCC
        after \~180-200 MeV;

    d.  S-TESLA section: FCCDCC or even FCCCDCCC.

> In the transitions between different sections the condition (1) should
> be satisfied. ~T0~ can not be higher than 90. Also if ~T0~, is lower
> than 30 the focusing is too weak. Therefore the "jumps" between the
> focusing period lengths should be less than \~2.5.

12. As a result of previous steps the linac structure is defined. Make
    TRACE3D calculations for phase advances, Twiss parameters as a
    periodical solutions and check transverse beam matching in the
    transitions from one to other focusing structure.

13. Longitudinal matching in the transitions. It may require some
    adiabatic change of voltages and synchronous phases. Depends from
    beam current. May require changing of the exact number of cavities.
    At this stage the studies can be done using the tracking code.
    TRACE3D can be helpful, requires a lot of hand-work to prepare
    appropriate input files.

14. Synchronous phases: typically --30 is OK in the low-frequency front
    end. Can be ramped to lower absolute value to the point of frequency
    transition. In high-frequency section it should definitely start at
    least --30 and gradually can drop to lower values. The longitudinal
    emittance can grow due to the rf jitter and the normalized
    longitudinal acceptance should increase with energy.

15. Definition of the synchronous phase: for the SC resonator: phase
    angle with respect to the rf phase that provides maximum energy
    gain. In SC resonators the true synchronous phase is equal to --90
    and particle motion occurs inside or outside separatrix with
    reference beta=~G~ of the given cavity type.

16. In the case of SC front end there is a specific problem to avoid any
    impact of inter-cryostat drift spaces on beam parameters.
    Transversely we do it usually by "missing" the cavity.
    Longitudinally it requires some careful synchronous phase setting in
    a number of neighboring cavities.

17. Final tracking with space charge, large number of particles, styduy
    possible sources of beam halo.

18. Make iterations of whole linac design to avoid any rms emittance
    growth in all planes. Reduce halo (or emittance of 99.99% of
    particles) if it is the case. The iterations can involve more
    careful beam matching in all sections, especially in the
    transitions.

19. More iterations may be needed to include some other considerations
    such as number of cavities, rf feeding, total length and etc.