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David M. Lee
Los Alamos
National Laboratory, Los Alamos, NM 87545
(submitted: 4 May 1995; revised: 15 May 1995)
The alignment requirements of the muon arm are considered. An error budget is
determined based on expected manufacturing tolerances, and expected alignment
tolerances. Simulation results are used as a guide.
Table of Contents
The alignment tolerances for the muon tracker are very stringent since Monte
Carlo simulations indicate that chamber resolutions approaching 100 microns are
required to get the good mass resolutions of the vector mesons. The alignment
of the muon system can be conveniently divided into two aspects, internal
alignment tolerances of the chambers in the muon arm (local coordinate system
(LCS)), and global alignment tolerances of the muon arm (global coordinate
system(GCS)) with respect to the rest of the PHENIX detector. The internal
alignment tolerances are the most stringent and will require very careful
attention to the mechanical construction of the chambers at each station and
the overall stability of the structural space frame holding the stations in the
lamp shade magnet. This note will address the alignment requirements of the
entire muon system so as to provide input to the mechanical design and assembly
procedures of the chambers and the design of the space frame as well as
providing the input required for the design of the active alignment system.
Gravity, thermal, and magnetic motions of the muon arm are considered as static
one time displacements that can be measured with optical and active alignment
methods to the accuracys discussed below and therefore will not be mentioned.
The only requirement is that the motions are within the range of the active
alignment systems. The local alignment system (LCS) will be considered as the
interior of the lampshade magnet with appropriate alignment fiducials. The
details have not been worked out. However, in the following discussion the
reference points will be the interior alignment fiducials.
The alignment requirements of the muon tracker involve specifications on the
initial placement of the muon chambers in the magnet, the stability of the
chamber systems after initial placement, the internal mechanical assembly of
the planes in the station, the station to station alignment tolerance, and the
alignment of the muon arm with respect to the rest of the PHENIX detector.
Physics requirements will drive the alignment tolerances but practical
considerations will define what is actually achievable. For this analysis the
results of physics simulations will be used as the primary guide.
The relevant coordinate system for the muon arm is r, 
, and z since the Muon Arm is intended to
measure the following characteristics of the detected particle,
 | transverse momentum by measuring the bend in the as a function of z; |
 | polar angle by measuring the radial coordinate, r, as a function of z; |
| azimuthal angle. |
However,
the specifications of the LCS will generally be given in terms of the chamber
measurement parameters, position along the anode wires ( high resolution and
approximately ) and anode wire
position (coarse resolution and approximately r). The Monte Carlo simulations
take these parameters as the input and determine the expected performance of
the muon arm for various physics parameters, i.e., mass resolution of the
upsilon. A position resolution along the anode wires of 100 microns is
desired. This corresponds to all contributions including the intrinsic chamber
resolution and all mechanical construction and alignment uncertainties.
Position resolutions in r of 0.5 cm and in z of 1mm are needed. The
contributions to the resolutions in either r or can be tabulated as follows.
 | intrinsic chamber resolution |
 | accuracy for mechanical assembly of the individual planes. |
 | alignment accuracy from station to station |
 | stability of each station |
 | initial placement accuracy of the octants |
The
Monte Carlo position resolution, 
, is
then
in phi
and
in r.
and
in z.
The
equation for phi implies that the intrinsic resolution of the chambers must be
less than 100 um or that the contribution from alignment and construction
errors will increase the resolution of the space point of beyond the intrinsic
resolution of the chamber.
The octant assemblies have been designed with three alignment pins on the top
and on the two sides of the frames. All external and internal alignment will
be determined with respect to these pins. The pin locations are fabricated to
a 25 um absolute tolerance. In the design and fabrication of the octant
assemblies we have tried to keep the distortions of the frames to as small as
possible. A finite element analysis of the support frame gives a maximum
distortion of the frame assembly of 30 um on the sides and 119 um on the top
when all of the foils and wires are stretched. We are planning on prestressing
the support structure to 30 um and 119 um when the etching and wiring is done
so the uncertainty in the position of the etched strips or wires with respect
to the true LCS will be comparable to optical alignment tolerances of 10 um.
The foil etching fixtures have manufacturers tolerance of 12.5 um on the lead
screw. The wire stretching jig has a measured uncertainty of 30 um from wire
to wire but a runout of approximately 200 um. The thickness of the octant
assemblies and the location of the individual planes will be limited to the
buildup of adhesive and the tolerance of the manufacture of the FR4 material
and it is estimated to be 200 um. Combining the azimuthal angle uncertainties
in quadrature (25,12.5,10 um), we expect the mechanical uncertainties in the
accuracys of the octant assemblies to be,
| azimuthal angle accuracy of the octant assemblies |  = | 29.7 um |
| radial accuracy of the octant assemblies |  | 200 um |
| z accuracy of the octant assemblies |  | 200 um |
Once
the octants are assembled we will assume that the assembly is a rigid
structure. Alignment fiducials on the space frame carefully surveyed with
respect to the alignment pins, will allow the octants to be aligned as a rigid
body to the reference points on the magnet.
The accuracy with which the stations will be aligned depends on the precision
of the active alignment monitors and their placement accuracy, and the short
and long term stability of the space frame holding the octants and stations
rigidly in place. Straightness monitors similar to those studied for use with
the GEM detector muon system have an accuracy of 5 - 10 um over a distance of
10 m and will be adequate for the PHENIX muon system. Stability of the octants
and space frame depend on the mechanical rigidity of the space frame as well as
the environmental modes induced into the magnet. These studies have not been
completed but we are trying to design the space frame to have uncorrected
motions due to environmental effects and mechanical rigidity to be less than 25
um in and r and .1mm in z. Cost
will more than likely define what the actual limit is. It is expected that the
alignment precision and the stability for both the r and coordinates will be the same. Based on
these numbers the alignment accuracys from station to station will be,
| azimuthal angle accuracy of the station to station alignment |  | 10 um |
| radial accuracy of the station to station alignment |  | 10 um |
| azimuthal angle stability of each station |  | 25 um |
| radial stability of each station |  | 25 um. |
| z accuracy from station to station |  | 0.1 mm. |
The initial placement of the muon chamber octants depends on the method of the
initial survey and on the range of the active alignment systems. For the
initial survey optical alignment requires lines of sight. Lines of sight at
the outer radius are possible, although difficult since we are designing the
chambers after the magnet has been `cast in concrete'. Lines of sight will
only be possible if provisions for lines of sight at the inner radius are
incorporated into the design of the octant support frame for station 2. Given
that station 2 is designed to allow the requisite number of lines of sight,
systems such as a theodilite with averaging to get rid of distortions due to
thermal air currents are required to give placement accuracys of 10 um. The
initial placement of the octants need only be within the range of the active
alignment system, ~ 5 mm, but a knowledge of where the octants are with respect
to the LCS must be within the optical alignment tolerance of 10 um in . In conjunction with the initial
placement by optical means the location of the octants will be determined by
the active alignment system. The initial placement precision is,
| azimuthal angle placement precision |  | 10 um |
| radial placement precision |  | 10 um |
| z placement precision |  | 10 um |
The
initial placement accuracy need only be in the range of the optical and active
alignment systems. We expect the placement accuracy to be,
| azimuthal angle placement accuracy |  | 5 mm |
| radial placement accuracy |  | 5 mm. |
| z placement accuracy |  | 5 mm. |
The following error budget is expected:
| Error | r | | z | |
| | 0.6 mm | 0.1 mm | | |
| | 0.2 mm | 0.03 mm | 0.2 mm | |
| | 0.01 mm | 0.01 mm | | |
| | 0.025 mm | 0.025 mm | 0.1mm | |
| | 0.010 mm | 0.010 mm | 0.010 mm | (precision) |
 | 5 mm | 5 mm | 5 mm | (accuracy) |
The
error budget for r and z are well within the tolerance required by simulations.
For combining the errors we get an
expected space point resolution of 115 um. To achieve a space point resolution
of 100 um, an intrinsic chamber resolution of 82 microns is required.
The accuracy of relating the muon arm local coordinate system (LCS) to the rest
of PHENIX is limited by the multiple scattering in the muon magnet steel and
the initial alignment of the detector subsystems to each other. Alignment
requirements of the North muon arm to the rest of the PHENIX detector, to the
south muon arm, and to the beam line will be done by comparing the results from
simulations. This takes into account the affects of multiple scattering and
the resolution capabilities of the muon arm.
For the purposes of aligning the muon arm to the rest of PHENIX, I have assumed
that the muon tracker can be considered as a rigid body.
Alignment to the beam line and vertex is important because errors in the
knowledge of these parameters translate directly into errors in the transverse
momentum. The alignment tolerance on the knowledge of the beam line and vertex
should be small when compared to errors that occur due to other fundamental
limits such as multiple scattering.
The accuracy of the radial coordinate in the global coordinate system (GCS) to
the LCS can be determined by the accuracy of the radial measurements in the
chambers. The resolution of the radial coordinate in the chambers is 1
cm/sqrt-12 or 2.9 mm. Simulations confirm that a 1 cm anode spacing is
adequate for preserving the mass resolution for the upsilon. As a "rule of
thumb" we will assume that we want the contributions to errors in any
coordinate to be a small fraction due to misalignment and will therefore be one
fourth of the measured coordinate. The radial alignment tolerance is then .75
mm.
Multiple scattering in the steel and copper nose cone can be used to find the
alignment tolerances in the azimuthal and polar coordinate as well as the
radial alignment tolerance at station 1 between the LCS and the GCS. The
energy of muons for which we want to determine multiple scattering should be
the highest energy muon. If we assume that these muons originate with the
decay of the 
than we will use an upper limit
of 10 GeV. Given this criteria the multiple scattering in the absorber
material is 12 mr and the angular uncertainty on the polar and azimuthal angle
is 7 mr. Projecting this to station 1 yields an uncertainty in r of 13 mm.
To keep the alignment errors in r small when compared to that from multiple
scattering we can use the 1/4 rule and define the alignment tolerance in r to
be 3.3 mm.
A 7 mr angular uncertainty on and
will also define the allowable uncertainty the
axis of the muon arm can be away from the beam axis, 1.75 mr from the 1/4 rule.
The length of the muon arm is 4500 mm so that the position of the end of the
muon arm with respect to the front of the muon arm should be known to 31.5 mm
in both x and y and applying the 1/4 rule, 8mm. This implies that we would like
to know the positions of each end of the muon arm magnet 8/sqrt-2 = 5.6 mm. in
each coordinate. Since we don't project back to a x or y coordinate in the
central magnet, the error in angular coordinate of the muon magnet with respect
to the true beam axis is the correct predictor of the allowed errors in the x
and y coordinate. Beam position monitors (BPM) can easily measure the location
of the colliding beams to much better than 5.6 mm. A possible location for
BPM's would be attached to the muon magnet back plate and the support structure
for the chambers at station 1.
The remaining global coordinate is z. Projecting from the spectrometer back
through the magnet steel and nose cone to determine the z coordinate will be
limited by the multiple scattering angle of 7 mr. Using the multiple
scattering error on the projection back to the vertex and using an average
angle of 20 degrees to project onto the z axis gives an error in z of 3.7 cm.
Simulations have shown that resolutions in z of the position of the vertex of
up to 2 cm have no effect on the mass resolution. The alignment tolerance in z
will then be one fourth of 2.0 cm or 5 mm.
The Muon Tracker alignment to the beam line and the vertex is,
| alignment tolerance in x an y |  and  | 3.7 mm |
| alignment tolerance in z | | 5 mm |
Comparing tracks in the Muon Tracker with tracks in the Muon ID is an important
method for rejecting pions in the Muon Arm. The alignment of the Muon Tracker
to the Muon ID must not degrade the ability of the track in the spectrometer to
point to the track in the Muon ID. The alignment of the muon tracker to the
Muon ID is limited by the multiple scattering in the muon magnet back plate and
first Muon ID plate and the smallest pad size in the Muon ID. The smallest pad
size is 1.0 x 2.8 degrees in theta and phi. The first Muon ID gap is at z=7250
mm so the pad dimensions are 12.7 cm x 35.5 cm. Multiple scattering from the
back plate (30 cm) is 6.6 mr for 10 GeV muons which gives a projected error in
x or y of 2.5 mm. This is small when compared to the pad size so the pad size
determines the alignment tolerance. An alignment tolerance of 3.2 cm is
sufficient.
The previous analysis of the alignment of the North Muon Tracker to the beam
line and vertex is basically the same for the South Muon Arm. However, to
consider the relationship of the South Muon Arm to the North Muon Arm the
invariant mass resolution of muon pairs can be used. The invariant mass of a
muon pair depends on the momenta of each particle, 
and 
, and the opening
angle, 
,
The
error on the missing mass is,
where
is the error on the opening angle and
= .
Assuming the momentum resolution to be 1.8 % then the angle term will dominate
for the smallest opening angle between the North and South Muon Arm, 110
degrees. To keep the error contribution from the angle term small we can apply
the 1/4 rule and equate the angle term to 1/4 of the momentum term and get = 13 mr. Since this is the difference between
the two muon angles the allowed tolerance on each of the muon angles is 
_ = 13/sqrt-2 = 9.2
mr. Since the 7 mr error that resulted from multiple scattering considerations
is more restrictive, it will be used to define the alignment requirements for
the South Muon Arm. The implication here is that aligning both muon arms to
the beam line and vertex is sufficient to insure good mass resolution.
A summary of the global alignment requirements are,
Alignment tolerance of North and South Muon Arm to the beam line and vertex,
Alignment
of Muon Arm to Muon ID
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