To date only ideal models have been exploited
to computationally describe the Fermilab Booster Synchrotron. Recent alignment
measurements of the Booster gradient magnets and comprehensive analysis
and compilations of these data have provided the basis for a more representative
picture of this accelerator. The purpose of this paper is to fully explore
the information contained in the alignment data and subsequently model
the Booster based on an "as built" or realistic description.
· Alignment Data
Magnet coordinates were obtained on almost all of the four alignment pins which mark each corner of a Booster gradient magnet, both F- and D-type, in the standard reference system used at Fermilab (DUSAF). Only a few pins on magnets were obscured, in particular by overhanging extraction components, and could not be surveyed. In some cases coordinates were obtained on the underside of these magnets which could be used in this analysis. Overall, the survey coordinates form a body of data from which a number of Booster machine parameters can be determined.
The upstream and downstream center of each magnet were obtained by averaging the appropriate alignment pins (A&B for upstream and C&D for downstream). The resulting upstream and downstream magnet center positions can then be used to determine the chord length between the two centers. Although averaged separately, the chord lengths of all F and all D magnets gave a consistent, overall length of 2.9144 m for both types of gradient magnets, to within survey errors (+/- .8 mm)1 and pin seating accuracy (+/- .8 mm)2. The measured magnetic arc length of the F (2.917 m), or D (2.916 m), as given in reference 3 extends beyond the alignment pin positions and also beyond the physical ends of the magnets. The bend angles of the F and D magnets were computed using their relative strengths and applying ring closure on the sum of the bends. The relative dipole strengths of the D and F magnets is quoted in reference 3 as:
Bo(D magnet)=Bo(F magnet) ¥ rF/rD.¥ LD/LF =.8538 ¥ Bo(F magnet)
where rF=41.411 m is the radius of curvature quoted3 for the F gradient magnet at 8 GeV, rD=48.488 m is the radius of curvature quoted3 for the D gradient magnet at 8 GeV, LD=2.916 m is the magnetic arc length of the D magnet, and LF=2.917 m is the magnetic arc length of the F magnet. With the relative strength parameter given above, and the fact that the total bend of the 48 D and 48 F magnets must sum to 2p, the bend angle of the F and D gradient magnets are .0707707354 and .0601289585 radians, respectively. Note that if bend angles derived from reference 3 are summed, (using the quoted magnetic arc lengths divided by the appropriate bending radius) the orbit does not close by about a degree. Knowing the bend angle allows the chord length derived from alignment pin positions to be superimposed across the traversal or magnetic arc length, and the difference in length to be calculated between the two resultant arcs. With this difference, the surveyed drift lengths can be corrected for extended magnetic fields, which is important for an accurate optical calculation. The resulting correction to an F magnetic end is +0.0010 m and to a D is +0.0006 m. The following table summarizes the Booster machine parameters as determined both from an average of all the survey data available for each parameter, and after applying the corrections discussed. (The column labeled 'MAGNETIC' incorporates the correction due to the additional magnetic field length.)
|(mag arc length/
*upstream-downstream chord between alignment pins
Once these machine parameters are fed into the computer simulation program, MAD4, a calculated closed orbit can be extracted and compared to the operational or "as built" closed orbit. To obtain an accurate operational description of the existing Booster, upstream and downstream radii were calculated for each magnet in a cell relative to an interpolated Booster center, and the eight data sets were subsequently averaged to find a mean radii for each of the specified cell locations. The true Booster center was determined simply by exploiting the symmetry of the ring. That is, the center position of a magnet's end was averaged with its reflected counterpart; the upstream center of the first D magnet in Cell 1 was averaged with the upstream position of the first magnet in Cell 13, and so on. In this fashion the Booster center was determined to be:
The table on the following page displays the calculated radii of interstitial cell positions including upstream and downstream magnet centers for an ideal Booster cell as compared to the actual radii measured for a representative Booster cell. Badly misaligned cells were not eliminated as they appear to cause less than a millimeter deviation in the overall averages. In fact deviations in the fourth column of less than a millimeter are not significant since a millimeter is approximately the rms value of the survey errors combined with the pin seating accuracy.
long straight drift
1st D magnet
1st D magnet
1st F magnet
1st F magnet
short straight drift
2nd F magnet
2nd F magnet
2nd D magnet
2nd D magnet
1.1 mm for and F-type and -1.2 mm for a D-type gradient magnet
Corrections to the previous table due to the position differences between magnetic endpoints and alignment pin positions provided far less than a millimeter correction and were thus neglected. The errors in the survey data are correlated since all position measurements are referenced to an absolute position or coordinate system; therefore, the radial position error is predominantly systematic rather than statistical, and is apparently on the order of +/-0.5 mm. Also, small changes in the magnetic corrections can easily produce changes on the order of +/- 2 mm in the calculated radial positions. Since errors on the measured magnetic characteristics of the Booster gradient magnets were not given in reference 3, an overall error of +/- 2 mm should be assumed in the calculated parameters due to this lack of information.
If one uses the above parameters and the bend angles based on the relative strengths of the gradient dipoles, then the basic Booster modeled with MAD gives the lattice description in the MAD table attached to this document. The markers SD, ED and SF, EF are the magnetic upstream and downstream centers of the D and F gradient magnets, respectively and Booster quadrants are displayed in the tables. The marker "begin Booster" denotes the center of a long-straight section and "end Booster" closes the orbit at the same point. One should note differences between the design Booster lattice as described in reference 3 (or, the second of the MAD tables presented here), and the as-built closed lattice. For example the long straight lattice parameters change somewhat,
bx = 6.012 ax = 0. Dx=-1.84 by= 20.007 ay =0. Dy=0.
bx = 6.493 ax = 0. Dx=-1.956 by= 19.786 ay =0. Dy=0.
Since the protons circulate counterclockwise in the Booster, here the dispersion has been corrected from reference 3 to reflect the correct negative value. (Historically, the Booster has been attributed with positive-sign dispersion implying the protons are circulating clockwise.)
A side benefit of the extensive analysis of alignment
data is that position errors can be assessed on a magnet to magnet basis.
The upstream and downstream deviations from ideal and averaged Booster
cell locations are given in the attached table. These alignment errors
have also been pursued extensively in the work by Shekhar Shukla5.
The format of his presentation of alignment errors has been adopted in
this paper in the figures following the table to provide a ready comparison
and, ultimately, validation of the "as built" Booster description presented
here. One should note that the magnet positions in the table have not been
corrected for the approximately millimeter offset between the optical and
magnetic center of the Booster magnets.
I wish to thank Jim Lackey and Shekhar Shukla for their useful discussions and essential archival documentation and information on the Booster Synchrotron design and implementation.
1. Alignment Dept., private communication
2. Jon Sauer, "Booster Optical Data Reduction," Fermilab, Mar. 19, 1973.
3. E. Hubbard, editor, "Booster Synchrotron," Fermilab publication TM405-0300, Jan., 1973.
4. Hans Grote and F. Christoph Iselin, "The MAD Program (Methodical Accelerator Design)," CERN publication CERN/SL-90-13(AP), version 8.4.
5. Shekhar Shukla, to be published in the "Proceedings
of the Division of Particle and Fields Meeting", Aug. 1994, Albuquerque.