[figure: Fermilab]

Doc Number: Beams-doc-4982 Version: 2.00 Category: Note

Aperture Implications for Recycler Assuming Full Coupling and Nominal Vertical Apertures

Bruce C. Brown Accelerator Division, Main Injector Department Fermi National Accelerator Laboratory ^{∗} P.O. Box 500 Batavia, Illinois

12 May 2016

Contents

1 Introduction 2

2 Recycler Characteristics 2 2.1 Vertical Beam Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Horizontal Beam Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Boundaries at Proposed Collimator Locations 6 3.1 Vertical Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Horizontal Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Observations and Conclusions 8

5 Further Observations 8

6 Acknowledgments 8

∗Operated by Fermi Research Alliance under contract with the U. S. Department of Energy

Abstract In order to specify the apertures needed for Recycler Collimators and Masks, we will explore the vertical aperture requirements implied by the nominal vertical beam pipe size and examine the lattice functions to find the largest vertical beta. For horizontal aperture requirements we will assume full vertical to horizontal coupling and add to this a requirement for the momentum spread and slip stacking offsets. To these requirements an allowance for injection errors will be added to arrive at an aperture requirement. Using a nominal beam emittance of 15 pi-mm-mr and the same momentum spread and slip stacking momentum requirements, we will determine the nominal beam size. The amplitudes required to move this beam edge to the aperture edge will set the maximum bumps needed for collimation. Since collimators will define apertures inside of the Recycler aperture, the bump amplitudes for collimation will be smaller than determined here. This is not a specification document but it will describe the basis we will use to define collimation system requirements.

1 Introduction

The Fermilab Recycler is now employed as a accumulation ring for 8 GeV protons to provide high intensity beam to inject into the Main Injector. With the available beam quality from the Booster, we have losses for all operating scenarios but studies of slip stacking indicate higher losses with these modes. To localize these losses, we are designing a collimation system which is expected to absorb protons which are outside the acceptance of the ring and the secondaries produced when those protons interact in large steel blocks. For designing this system we need to know the boundaries for beam which is transmitted through the limited vertical aperture of the Recycler, and compare that aperture to the expected beam size in order to understand the magnitude of potential orbit bumps required to move beam into the collimation system. We will use a nominal Recycler lattice and a tentative collimator placement for which we will document the important geometry.

2 Recycler Characteristics

The Courant-Snyder lattice parameters we will employ come from the R90 Console Program. We employed R90 file 16 from 8 September 2015 to obtain a lattice description. The lattice included preliminary descriptions for a number of potential primary and secondary collimator locations and many potential locations for dipole corrector magnets. Using the text output from R90 we loaded this description into an Excel spreadsheet (RRApertureForBeam.xlxs) to calculate beam properties. This ‘design’ lattice has been compared with measurements and found to be in reasonable agreement. We choose not to add an allowance for the difference between ‘design’ and ‘measured’ lattice to our already conservative assumptions below.

2.1 Vertical Beam Size

We assume that the principal limitation is in the vertical acceptance due to the standard beam pipe height. This will manifest itself at the largest β_{y} around the ring. In this lattice file we find β_{peak} = 58.6 m. For the maximum aperture we use y_{pipe} = 20 mm. The internal beam pipe height at the center is a bit more than 22 mm but there are many welds at high β_{y} so we assume that this should cause us to assume a smaller maximum available size. The vertical boundaries will then be

y_{max}(s) = \begin{matrix}√ & \overline{β(s)} \\ √ & \overline{β_{peak}} \end{matrix}y_{pipe}  (1)

Figure 1: Vertical beam properties in the region from RR611 to RR620 are shown. The orange curves mark the upper and lower vertical boundaries of beam acceptance. Beams which can circulate through the vertical aperture limit at the maximum beta will fit inside of these boundaries after correcting injection offsets. The blue curves describe the beam envelope for 15 pi-mm-mr beam. Red with green triangles mark positions under consideration for collimation. The horizontal axis (STATION) is the distance from the MI10 injection marker (in meters).

y_{min}(s) = −\begin{matrix}√ & \overline{β(s)} \\ √ & \overline{β_{peak}} \end{matrix}y_{pipe}  (2)

Using typical Booster 95% normalized beam emittance of ϵ = 15 pi-mm-mr we calculate typical RMS vertical beam sizes from

σ_{y}(s) = \sqrt{\frac{ϵβ_{y}(s)}{6 β_{rel}γ_{rel}}}  (3)

where β_{rel} and γ_{rel} describe the proton relativistic motion. From this we find a three sigma boundary

y_{up}(s) = 3σ_{y}(s)  (4)
y_{down}(s) = −3σ_{y}(s)  (5)

For the region from RR611 to RR620 where the collimators and collimator bumps are planned, we show the typical beam size and boundaries of accepted beam in Figure 1. Locations under consideration for collimation are identified with green triangles with red outlines. These include a primary identified in the spreadsheet as PCOLL613B and Secondary Collimators identified as SCOLL613D and SCOLL616 (for initial phase of collimation - 2016) and potential additional Secondary collimators (Alt Sec - potential second phase installation) at SCOLL614 and SCOLL619U.

Figure 2: Horizontal beam properties in the region from RR611 to RR620 are shown. Assuming full coupling, beam which circulates through the vertical aperture limit at the maximum vertical beta will fit within the orange curves after correcting injection offsets. Blue curves show the beam size for 15 pi-mm-mr beam with the momentum width allowed by slip stacking acceptance. The green curves show the same beam which has been displaced for slip stack slipping. The x offset curve shows the displacement of the beam center for the slipping beam (momentum orbit for slipping beam). We see that the edge of the on momentum beam is just beyond the center (offset beam position) of the slipping beam. We are using the convention that x positive is radially out (not the convention used in R90).

2.2 Horizontal Beam Size

For each injection from the Booster (at 15 Hz), the injected beam will circulate for about 6000 turns before the next injection. We assume (round beam) that some combination of linear (skew quadrupole) and non-linear coupling will mix all the beam vertical and horizontal motion and especially so for the beam at the boundaries. This assumption implies that we calculate the horizontal betatron beam size with the same parameters as for the vertical beam size.

To this we must add a contribution for momentum effects. Two contributions are of interest. The longitudinal admittance for 15 Hz slip stacking contributes an RMS momentum spread of 3.5 MeV/c for injected beam which can be accepted. The 20 Hz Booster operation may allow a larger admittance which we will take as 4/3 larger or 4.667 MeV/c. For each of the slip stacked beams we will calculate a horizontal beam sigma which is the RMS of the betatron RMS and and momentum RMS. For an 8 GeV or 8.889 GeV/c beam these imply \frac{σ_{p}}{p} of 0.000394 (15 Hz) or 0.000525 (20 Hz).

σ_{x}(s) = \sqrt{\frac{ϵβ_{x}(s)}{6 β_{rel}γ_{rel}} + (1000 η\frac{σ_{p}}{p} )^{2}}  (6)

The factor of 1000 is because we express beam sizes in mm. We also comment here that η is negative in this lattice. Before calculating the positions of the beam edges, we will calculate the displacement of the beam caused by the momentum offset for slip stacking. This is determined by the time available for slipping and is dp = 24.5 MeV/c (dp/p = 0.276%) for 15 Hz Slip Stacking or dp = 32.66 MeV/c (dp/p = 0.367%) for 20 Hz Slip Stacking.

xssoffset = 1000 η \frac{dp}{p}  (7)

The 3 sigma edges for the beam at momentum center are given by

x_{in}(s) = 3σ_{x}(s)  (8)
x_{out}(s) = −3σ_{x}(s)  (9)

For the beam which is decelerated for slipping, the edges are

xoff-in(s) = xssoffset+ 3σ_{x}(s)  (10)
xoff-out(s) = xssoffset− 3σ_{x}(s)  (11)

For the maximum horizontal beam size we will be more conservative by adding the momentum width, momentum offset, and betatron sizes linearly. We seek to know the size we need to leave when we build new devices. Slip stacking may employ beam on momentum center along with decelerated beam (Spring 2015 option) or on center with accelerated beam (possible future option) or we may inject off center and displace the slipping beam from that orbit. We will never use the aperture we describe with this calculation but it will allow full flexibility for future decisions.

x_{max}(s) = \begin{matrix}√ & \overline{β(s)} \\ √ & \overline{β_{peak}} \end{matrix}y_{pipe} − 1000 η (dp + σ_{p})  (12)

x_{min}(s) = −\begin{matrix}√ & \overline{β(s)} \\ √ & \overline{β_{peak}} \end{matrix}y_{pipe} + 1000 η (dp + σ_{p})  (13)

The horizontal beam features are illustrated in Figure 2. Again we show typical beam size and boundaries of accepted beam but the beam boundaries are shown both for on momentum beam and beam decelerated for slipping. Also shown as (x offset) is the beam center (momentum orbit) for the decelerated beam

3 Boundaries at Proposed Collimator Locations

With the above formulas implemented in the spreadsheet RRApertureForBeam.xlxs, we can find the expected beam sizes and maximum beam sizes for the proposed collimator locations. will provide the spreadsheet with momentum parameters for 20 Hz slip stacking but for 15 Hz values, the user can substitute in the two cells as desired.

3.1 Vertical Boundaries

For the vertical boundaries, we can put everything in Table 1. The position of the largest vertical displacement we can require at each location is y min or y max. We see that this is a bit smaller than the 20 mm size we assume for the beam pipe inside aperture. We suggest that an allowance for injection steering errors of up to 2 mm should be added to this. We can consider whether the appropriate vertical size of a collimator should be uniform or whether smaller collimation apertures corresponding to y max as small at 9.37 mm might permit a reduced external activation.

For designing bumps, we see that the allowed displacement (y max - y up) also varies by x2 and while that does go with the Beta at the collimator, one needs to design specific bumps to see what magnet strength is needed.

Table 1: Vertical Lattice and Beam Boundaries and Maximum Bump Amplitude

Name  Station phase Beta  Alpha  y dn  y up y min y max y max - y up
m  Rad/2π m  mm  mm  mm  mm  mm
PCOLL613B 2853  20.93 50.746 0.04275 -10.978 10.978 -18.612 18.612  7.633
SCOLL613D 2861  20.96 31.782 1.7997 -8.688 8.688 -14.729 14.729  6.041
SCOLL614 2878.1 21.12 25.12 -1.65082 -7.724 7.724 -13.095 13.095  5.371
SCOLL616 2906.7 21.34 12.849 -0.9175 -5.524 5.524 -9.3652 9.3652  3.841
SCOLL619U 2939.8 21.66 57.391 -0.04248 -11.675 11.675 -19.793 19.793  8.118

3.2 Horizontal Boundaries

We need to display the horizontal information in more tables to keep them of manageable size. For finite beam momentum spread, the dispersion (η or Eta) increases the beam size. But more importantly, we will be working with slip stacking momentum offsets of one of the beams. We consider that in one direction and exhibit the required displacements. The vertical collimation will impact both beams. With the preferred two secondary collimator solution, we will seek to collimate the on momentum beam. Only by adding at least one of the alternative collimators will it be possible to provide horizontal scraping for the offset (slipping) beam. Perhaps the collimator vertical boundaries will be sufficient to control beam halo. In Table 2 we show the lattice parameters with the RMS beam size and the slip stacking displacement.

Table 3 shows the boundaries for 15 pi-mm-mr beams for both the beam displaced for slipping and the beam on momentum center (as injected). Also shown is the maximum beam extent due to betatron beam size, momentum beam size and momentum displacement for slipping (20 Hz slip stacking) added linearly. We see that there is significant overlap between the 3 sigma widths of the displaced and centered beam. Again, we suggest adding 2 mm to allow for injection errors.

Table 2: Horizontal Lattice, beam size, and offset for 20 Hz Slip Stacking

Name  Station phase Beta Alpha Eta  Etap sigma x offset
m  Rad/2π m  m  mm  mm
PCOLL613B 2853 21.797 11.514 -0.004 -1.131 0.0006 1.841 -4.154
SCOLL613D 2861 21.885 24.034 -1.630 -1.478 -0.07923 2.635 -5.431
SCOLL614 2878.1 21.949 34.967 2.0269 -1.667 0.08288 3.161 -6.124
SCOLL616 2906.7 22.164 22.163 2.1675 -1.246 0.12183 2.505 -4.579
SCOLL619U 2939.8 22.530 12.372 0.0459 0.003 -0.00354 1.807 0.011

Table 3: Horizontal Boundaries for 20 Hz Slip Stacking

Name  Station x off out x off in x out x in x min x max
m mm mm mm mm mm mm
PCOLL613B 2853 -9.679 1.367 -5.524 5.524 -13.613 13.613
SCOLL613D 2861 -13.337 2.475 -7.906 7.906 -19.016 19.016
SCOLL614 2878.1 -15.608 3.359 -9.484 9.484 -22.449 22.449
SCOLL616 2906.7 -12.095 2.937 -7.516 7.516 -17.533 17.533
SCOLL619U 2939.8 -5.410 5.432 -5.421 5.421 -9.1772 9.1772

Table 4: Horizontal Displacement Requirements

Results for 20 Hz  Results for 15 Hz
x max - x in -x min + x off out x max - x off out -x min + x off out
mm  mm  mm  mm
PCOLL613B  8.089  3.935  7.030  3.913
SCOLL613D  11.110  5.679  9.710  5.636
SCOLL614  12.965  6.841  11.377  6.782
SCOLL616  10.017  5.438  8.823  5.387
SCOLL619U  3.757  3.767  3.760  3.768

4 Observations and Conclusions

The primary collimator should have a width of at least 2 × 13.6 mm (See Table 3). Note that we will bump the beam into this after damping and do not need to add an allowance for injection errors. The horizontal beam position at the primary is unconstrained. But we observe that changes in the horizontal beam position at the first secondary (SCOLL613D) will necessarily change the horizontal position at the primary. These position changes fall within the limits for the primary size specified above.

The secondary collimators for the Main Injector employed an aperture of 2 inches by 4 inches. This is over generous. We used 2 inches by 2 inches in the MI8 collimators and this looks to be adequate for the Recycler Collimation. The required minimum size (with these assumptions) can be determined from (y max)-(y min) in Table 1 and (x max)-(x min) in Table 3 and that is in some cases much smaller. The motion provided for the secondary collimators will accommodate larger collimator apertures. Perhaps we can learn how much advantage a smaller aperture provides when seeking to contain the activation within the secondary collimators by employing MARS calculations.

5 Further Observations

Version 1 of this document (October 2015) had a couple of significant errors which required a new version. The beam size was miscalculated in the spreadsheet by \sqrt{π} with the result that the plots suggested much more free aperture than was real. Errors in Equations 3 and 6 were corrected. The specification for collimator apertures is available currently in the RecyclerCollimation web page. It will be desirable to provide a new document with this information. The viewpoint of this document was employed for that work but additional allowances for uncertainties were included, making the apertures larger.

6 Acknowledgments

I would like to thank Ming-Jen Yang and Phil Adamson for many discussions and suggestions.