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Observations on the NUMI proton beam
David C. Carey
Fermilab
13 Dec 2001
The purpose of the NUMI proton beam line is to transport protons from the main injector to the NUMI target. The distance is approximately 350 meters.
The NUMI beam leaves the MI 60 straight section in the main injector and travels northwest toward Soudan, Minnesota. It drifts a couple of hundred feet and crosses over the antiprotons from the main injector. Finally it comes to the NUMI stub, where it is bent down 157.8 mr.
The target hall is in dolomite bedrock 20 m below. Between the MI level and the bedrock is glacial till, which is loose and has to be mined by hand. A tunnel through it has to be strong enough to support itself in this loose material. For this reason the tunnel is expensive and needs to be made as short as possible. In the pretarget hall the beam is bent up so that it is aimed at Soudan. It is still going down at an angle of Ц58 mr. Finally the beam strikes the NUMI target in the target hall.
The only other bends of significance are associated with the main injector. At the extraction point are located a set of Lambertsons, which give the protons a vertical kick to get them out of the accelerator. About 30 m downstream is a set of bending magnets which perform a dual function. They bend the protons down so as to level them out, and they bend the proton beam horizontally away from the injector.
The standard method for transporting a beam of particles a large distance is by means of a periodic system. A periodic system can take many forms. The most common of these is the FODO where quadrupoles are uniformly spaced and alternated in polarity.
In the first design of Lucas and Childress, NUMI proton beam started out as a FODO. However, the constraint of a tight fit in five variables (betax, alphax, betay, alphay, and etay -- all at the end) affects the five upstream quadrupoles and results in values in the thousands for the two beta functions. Doublets were reversed in polarity by hand and the system was tried again. Sometimes the process of fitting would reverse a doublet.
Various versions of the beam line were tried. There were two that were most significant. They came to be known by the names of more_quads (mq) and better_focus (bf). The mq version has two additional quads in the carrier tunnel region. The bf version is a better focus but not as compared to the mq version. The bf version produces a better focus than another version also with no quads in the carrier tunnel region.
The part of both versions before the carrier tunnel region has four quadrupole doublets. The mq version has another doublet in the carrier tunnel region. Both versions have optical triplets in the pretarget hall. However the triplet is formed with three physical magnets in the bf version and with four magnets in the mq version.
Portions of these two beam lines look like they were part of a periodic system, or had some periodic structure to them. The first two sets of four quads in the mq lattice before the carrier tunnel region each form a symmetric optical triplet. The second set of four quads in the bf lattice has the same layout. From the distance to the first focus, or from the direct calculation of the phase shift versus the accumulated distance, the phase shift per cell can be estimated to be in the vicinity of 90 degrees. However, none of the structures described is part of a larger periodic system.
There are three levels of specificity that are commonly used in a fitting procedure. The first is block diagonalization. Here the correspondence of the varied parameters with the constraints is rather sharply delineated. The beam line is segmented longitudinally (corresponding to the block diagonalization) and the varied parameters and constraints that affect them are located in the same segment.
The second method is sometimes called all-at-once. Here the correspondence between the varied parameters and the constraints is not so specific. All-at-once is often used when the imposed constraints represent required conditions distributed throughout the beamline. One example is the NUMI proton beam where the beam is made to be as small as possible by fitting the beta functions in the two transverse planes at various places.
This Уhybrid fittingФ can be dangerous if the computer determines a correspondence between a varied parameter and a constraint that was not intended by the user. However, Уall at onceФ, when successful, can be a good indicator of the way to go to achieve the desired constraints. Lucas has made good use of the Уall-at-onceФ method and has thereby determined some useful configurations.
The third useful method is massaging. In this method one can move elements a short distance or interchange adjacent elements. One can also change parameters one or a few at a time and by slight amounts. It is a form of exploration. My impression is that Lucas et al. relied heavily on this method. Massaging is a valuable method as it allows the user to explore the functional dependences that might not be obvious in a multidimensional space. Lucas interchanged elements one at a time and moved then by slight amounts to arrive at the best configuration. He has arrived at a solution where the beam size is comfortably smaller than the apertures.
I consider the solution to be perfectly satisfactory. However, one can also consider an alternative approach and see where it leads. As we go through the beam line, I think we shall see that we are not so far from the approach Lucas used.
A quadruplet of quadrupoles can be configured so that the transfer matrix is the negative of the identity matrix. A repetition of the same structure leads to a transfer matrix which is the identity matrix. Variations can be found which magnify or demagnify in the transverse planes.
If the polarities of the quadrupoles are reversed in a periodic system, the system will then necessarily have transfer matrix elements which are off-diagonal. Such transfer matrix elements, if uncontrolled, can grow to be quite large. There is an advantage to using a design where the off-diagonal matrix elements are eliminated by the symmetry of the configuration. A configuration where the off-diagonal elements are all equal to zero is called a telescopic system.
The NUMI hadron beam could then consist of several sections. The first would ordinarily match the phase space of the main injector into the periodic system of the NUMI beam. Since the NUMI beam is yet undetermined, the matching configuration must be considered as a single problem combined with the rest of the beam line, with coupling between the various parts.
The second section would be a magnifying telescopic system. It would make the beam phase space large with small angular divergence. The purpose would be to span the carrier tunnel region.
The third would be a demagnifying section. It would restore the phase space configuration that existed at the beginning of the second section. This would include the width in two transverse planes and the two angular divergences.
We temporarily ignore the fact that there is no space for the final telescopic system. If there were room for it, its purpose would be to focus the two betas, two alphas, and the vertical eta at the NUMI target. The number of constrained quantities is greater than the number of independent magnifications. The matching section of the beam line may supply a sufficient number of degrees of freedom to fit the alphas
All of this description is very rapidly coming to have a strong resemblance to LucasТ beam line. Another year and a half of massaging the data could produce a solution. As for now, what we have described is a hypothetical variation. The variable parameters which control the final focus are rapidly propagating upstream in the beam line.
If the beam line were recombined, the value of etay need not be constrained by the final focusing section. The dispersion is calculated as the integral of the 3/2 power of the vertical beta function times the sine of the phase shift times the curvature. The integral is taken with respect to the phase shift and multiplied by the magnification. For the derivative of the dispersion the integral involves the cosine of the phase shift and is multiplied by the angular magnification. Since the two bends are in the opposite direction, they need to be at least 90 degrees apart for there to be any cancellation.
For the bf version of the beam line, the entire length is used to accumulate a phase shift of 360 degrees. Since the sign of the sine of the phase shift is the same in the two bending sections, there is some cancellation. By appropriate motion and retuning of quadrupoles, it may be possible to maximize the cancellation and thereby reduce the number of degrees of freedom that the final focus must deal with.
For the mq version of the beam line, the accumulated phase shift becomes 360 degrees at the point where the quadrupoles in the carrier region are located. This means that the sine of the phase shift has opposite signs at the two vertical bend strings. There is no cancellation and the final set of quadrupoles must do all the focusing. This is a case where the placement of a quadrupole in the carrier tunnel actually makes the situation less convenient.
Lucas has a graphing feature in his Excel-based version which I do not have in my Top Drawer based model. That is the ability to plot the results of misalignments. He has showed me his results for tracing the displacement of a beam due to each possible displacement of a magnetic element.
From the results of this misalignment graph Lucas has described to me a situation where a 1 mm transverse translation of a quadrupole will produce a 24 mm displacement of the beam centroid at a later point. Such a magnification effect arises when the quadrupoles in a cell in a periodic system are reversed in polarity. The instability comes from the fact that there are two or more defocusing quadrupoles immediately before the location where the large displacement shows up. The instability often does not show up as readily in the beam line as a whole. The reason it does not is that in the beam line as a whole, the two defocusing quadrupoles may be preceded and followed by focusing quadrupoles.
A disadvantage of the use of telescopic systems is that it uses larger units for construction of a charged-particle optical system. Larger units means that there will be fewer independent units and the system may not be as flexible, and therefore it would be more difficult or impossible to produce a solution.
With thanks to Peter Lucas and Sam Childress
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