The cell lengths in linacs of conventional design
are typically graded as a function of particle velocity. Use of
symmetric cells in short segments of both the coupled-cavity drift-tube
linac (CCDTL) and coupled-cavity linac (CCL) simplifies the cavity
design. Mechanical design and fabrication are also simpler without
compromising the performance. We have implemented a design algorithm
in the PARMILA code for symmetric cells and symmetric multiple-cavity
segments. This feature significantly reduces the number of unique
components. We have compared the performance of a symmetric-segment
linac with a more conventional graded-cell-length linac.
Figure 1 defines a cell as an acceleration unit that includes a single rf accelerating gap. A cavity contains one or more accelerating cells and may be resonantly coupled to other cavities.
CCDTL [1] cavities may contain two types of cells. One type extends from the center of one drift tube to the center of the next. The other type starts from the up-stream face of a cavity and extends to the center of the first drift tube. The reverse of this later type occurs in the last cell of a CCDTL cavity. In a graded- design, each cell in a CCDTL would have a unique length and the resulting cavities would be asymmetric. In a CCL, cells and cavities are synonymous and are typically symmetric.
We define a symmetric unit or segment as a series of coupled cavities all of which by themselves are symmetric and all of which have identical geometry. In symmetric units the rf fields in each cavity are equal. In the CCDTL, the field amplitude in each cell may be different but is fixed relative to the other cells within the cavity. In a symmetric unit, symmetrically placed cells have equal fields.
Symmetric units may include space between cavities
for focusing elements. Symmetry does not constrain the placement
of the focusing elements. Ignoring the quadrupole lenses, a symmetric
unit is non-directional. It will perform correctly with either
end located up-stream. The quadrupoles position is independent
of the cavity position so long as it fits within the space between
the cavities.
In a graded- linac design ( = relativistic particle velocity) a synchronous particle determines the cell length by requiring it to arrive at the center of the accelerating gap when the rf fields are at the "synchronous" phase. Any phase programming along the linac is folded into the increasing cell lengths which are otherwise proportional to . ( = the rf wavelength.) In PARMILA [2] the thin-lens approximation determines the acceleration across a gap [3]. Longitudinally, the Prome term [4] corrects the phase advance across the gap ensuring that thin-lens approximation conserves emittance.
PARMILA divides the gap at its electrical center zc. In a multiple-drift-tube CCDTL cavity, the fields in the end cells are asymmetric and the geometrical and electrical centers do not coincide. Therefore, we use the code CDTFISH [5] to design the CCDTL cavities so that the electrical center of the gap coincides with the center of the gap as defined by PARMILA. The value of zc, can be found by satisfying the equation:
. (1)
CDTFISH adjusts the length of the cavity noses until the electric field integrals in equation (1) are equal for a given value of zc. It does this while maintaining its resonant frequency. For each geometrical velocity g SUPERFISH [6] then calculates the transit time factors and other relevant cavity parameters. This procedure is now used by PARMILA to determine the cell lengths of a graded- linac. In principle, a linac designed by a graded- method accelerates slightly more efficiently than one designed by the symmetric method.
To maintain an average phase synchronism over a symmetric unit, PARMILA sets the entry phase to the symmetric unit so that average phase equals the design phase. The average phase is the phase, averaged over all gaps in the symmetric unit, of the reference particle when it arrives at the gap centers. In conventional graded- designs, the reference particle sees cells of increasing length. However, imposing equal cell lengths, optimized near the mid section of the symmetric unit, inevitably results in cells that are the incorrect length at both ends of the symmetric unit. A reference particle sees a longer cell length than preferred in the earlier cells, and a shorter cell length in the later cells. This results in a phase slip, shown in Fig. 2 through the symmetric unit. We determine g so that the reference particle phase increases at successive gaps until the mid section, then, decreases from above the design phase until the end of a symmetric unit.
In designing a symmetric linac, we define a "design particle" and a "reference particle." We no longer use a "synchronous particle" that arrives at the center of cells at the synchronous phase. The purpose of the loosely-defined design particle is to keep track of the phase programming. The design phase may not correspond to phase at any particular gap for multiple-gap symmetric unit.
The reference particle is a sample particle that obeys the particle dynamics. PARMILA chooses the length of the symmetric unit such that all the phases seen by the reference particle at the gap centers are close to the design phase. We determine the geometrical velocity g from the requirement that the time required for the reference particle to traverse a symmetric unit is equal to a time that should lapse to maintain synchronism at both ends of the unit. We impose no restrictions on the phase or energy of the particle at the gaps within the unit. The total length of a symmetric unit must be nsg/2 (ns is an integer). The geometrically determined velocity g is constant over the symmetric unit. We have investigated alternate approaches for determining g such as choosing its value corresponding to half the energy gain in the segment, or to half the velocity gain in the segment. These schemes were unsatisfactory, particularly when the number of cells in a symmetric unit was small.
In our calculation of g we include the Prome-phase correction [4] as an extra time contribution in a gap transformation. Previously PARMILA used this only in the particle-dynamics portion of the calculation. The effect of a positive Prome value is a reduction in the cell lengths. For example, for the 2-drift-tube CCDTL in Fig. 1, The following expression determines bg.
(2)
where T is the rf period (/c). ji and jo are the relativistic velocities of entry to and exit from the gap j. Terms Pj (j=1,2,3) are the time delays derived from the Prome phase corrections for each gap (Pj = /(2c) (phase correction)j .) With this correction, the cell lengths and the particle dynamics through the symmetric unit become consistent. An error the order of 0.1 degrees per cell accumulates during the cell generation process when the Prome correction is neglected.
After determining the symmetric-unit length, PARMILA adjusts its longitudinal position so that the reference particle arrives at the entrance to the symmetric unit at the correct phase. The exit phase of the reference particle traversing a symmetric unit differs from its entrance phase to the next symmetric unit. Also the exit phase of a reference particle through a symmetric unit does not necessarily equal its entrance phase. PARMILA adjusts the entry phase to minimize this phase difference using single particle dynamics through each symmetric unit. After determining the entry phase, the code calculates the length of the drift space between the units to provide the exit to entry phase difference. If the design requires a ramp in the phase, adjusting the drift-space length provides the ramp. If external-quadrupole magnets require additional space, it is added in units of /2 to maintain the correct reference particle phasing.
If the linac design requires a ramp in the accelerating
field gradient, we step E0, the average axial electric
field, from unit to unit while maintaining E0.constant
within the unit. For a given ramp in E0, the value
of g is required to calculate the needed value of E0.
Therefore, we iterate about 5 cycles through each symmetric unit
until the correct g is found for the E0
calculation.
The average phase of reference particle must bear some relation with the design particle phase. Because the design particle is simply a programmed phase angle, there are a number of ways to approach this problem. Using a reference particle that follows the correct beam dynamics, one can constrain the entry and exit phases of the reference particle through the symmetric unit to be equal. However, if the number of cells in a symmetric unit is small (2 or 3), we often encounter a difficulty. The extreme phases (phase at gap 1 through 4 in Fig. 2) do not bracket the design phase. Another approach requires the equal phase angles at the center of first and the last gap in the unit as seen by the reference particle. This approach also suffers from the same problem.
We therefore require that the average of reference particle phases at gap centers in a symmetric unit equal the design phase. This approach ensures the extreme phases in the symmetric unit bracket the design phase. The entry and exit phases are not necessarily equal. In the PARMILA code, we employ Brent's zero-crossing technique [7] in determining the entry phase angle. For each iteration, a single particle is propagated through the symmetric unit. This method converges quickly and accurately.
When the number of gaps becomes large in a symmetric unit, g is appreciably different from the actual particle near both ends of the segment. This situation requires a correction in the transit time factors for the reference particle. We use CDTFISH to design the cavity for specified g. Then SUPERFISH calculates the transit time factors (T, T, S, S, etc.) assuming that the reference particle velocity =g. When the reference particle is not equal to g, we expand the transit time factors around g with respect to the wave number k (k is 2/). PARMILA uses this expansion in for both the single-particle-dynamics calculations in the linac design and in the multiple-particle simulation sections of the code.
After determining the entry phase, PARMILA stores the phases at the entrance, center and exit of each cell as well as the energy at the exit of the unit. Normally, the exit phase of one unit is not equal to the entry phase of next one. The source of this can be partly a ramp in the phase, and partly the phase slip through the multiple cells in the unit. Adding extra space between units corrects this small phase discontinuity. The PARMILA code generates one additional symmetric unit beyond the end of linac. It uses this additional unit to calculate the correct spacing between the end of this section of linac and the next section if more follows. The design of the next unit uses the exit energy of this unit as its starting energy. The linac is designed by repeating this process until the required energy is achieved. All of the pertinent design information is stored in memory for the multiple-particle beam-dynamics simulation.
In a more general CCDTL structure (see Fig. 1, CCDTL
with 2 drift tubes), the cell lengths within a cavity may differ
according to their position in the cavity. Multiple cavities that
are completely interchangeable may comprise symmetric units. In
a symmetric unit, if there are no external quadrupoles between
cavities, either end can be placed to the beam up-stream. If there
are multiple drifts within a symmetric unit for quadrupoles, the
unit may not be reversible because the drift lengths may be different.
Each symmetrically designed unit has a unique g and
all of the cell lengths within the segment are proportional to
g.
We designed a 2-drift-tube CCDTL that accelerates
a 100-mA proton beam from 8 to 20 MeV in two ways; The conventional
graded- approach and the symmetric unit approach. Table 1 compares
the two of designs at the end of 59 cavities.
| Length
(cm) | Energy
(MeV) | # of
cavities | #of
cells | |
| Graded- | 1639.9 | 20.274 | 59 | 177 |
| Symmetric | 1637.9 | 20.215 | 59 | 177 |
These designs use a ramp in both the cavity field
amplitude and the synchronous phase: The cavity phase varies from
-54 to -40 and the field E0 varies from 1.56 MV/m to 2.26 MV/m.
The two designs differ by 0.06 MeV and about 2 cm in length.
The new version of PARMILA can design symmetric cavities
and symmetric linac sections. It does this for both CCDTLs and
CCLs. Then it calculates the beam dynamics through the linac.
The symmetric design process simplifies the engineering and eases
the fabrication. The difference in performance of the graded design
and the symmetric design of a CCDTL is small.
Work is supported by the US Department of Energy.
[1] J. H. Billen et al., "A New Structure for Intermediate-Velocity Particles," Proceedings of the 17th International Linac Conference, Tsukuba Japan (August, 1994).
[2] H. Takeda et al., "Modified PARMILA code for new Accelerating Structures," Proceedings of the 1995 Particle Accelerator conference, Dallas, Texas, May 1995.
[3] Linear Accelerators, P. M. Lapostolle and A. L. Septier, editors, North Holland Publishing Company, p 759 (1970).
[4] Michael Prome, "Effects of the space charge in proton linear Accelerators," LA-TR-79-33, p 37-39 (1979).
[5] J. H. Billen et al., "POISSON SUPERFISH," LA-UR-96-1834, P 217, May 1996.
[6] J. H. Billen et al., POISSON/SUPERFISH on PC Compatibles," Proceedings of the 1993 Particle Accelerator Conference Vol. 2 of 5, p 790-792.
[7] W. H. Press, Numerical Recipes in C, Second Ed. (Cambridge, 1992), p 359.