Calculating the move

Calculating Position of Each Bundle Support Mounting Rod

An Excel spreadsheet has been created to calculate the location of the bundle supports' mounting rods with respect to the focal plane coordinate system. This spreadsheet also calculates the length of the parallel railing end supports which need to be fabricated anew for each unique TAGM location on the focal plane. One last, but very important thing that the spreadsheet calculates is the shim size needed, during TAGM realignment, in order to achieve the proper tow angle between bundle supports during mounting. In addition to the spreadsheet, an AutoCAD drawing used to design the three parallel rail components needed for each unique tagging energy spectrum starting position of the TAGM and a drawing for the current setup (β = 12.5^o to 11.06^o). These files are in US standard units (inches) and to scale with a tolerance of ± 0.001 inch.

A summary of the spreadsheet calculations is a follows:

Figure 1: Sketch showing the X-displacement along the focal plane (Δx) from the center of the first fiber column to the center of the sixth fiber column of the first bundle support.

• Select a starting energy for the photon tagging array (highest γ energy to tag)
• Using hodoscope energy bin bounds interpolate the crossing angle with respect to the focal plane (β₁) of an electron associated the highest energy to be tagged (E_γo)
• Interpolate the location on the X_FP axis at which this electron crosses (X₁)
• These electrons will pass through the center of the first column of SciFi fibers
• Using β₁ calculate the X-displacement along the focal plane (Δx) from the center of the first fiber column to the center of the sixth fiber column of the first bundle support
• Add Δx and X₁ to get X₆, then interpolate the value of β₆
• Using the average value of β₁ and β₆ (β_avg.), recalculate the X_FP displacement (Δx) from the center of the first to sixth fiber column (X₆)
• Repeat the above two steps until the bundle support angle β (e.g. average between β₁ & β₆) does not change appreciably

Now we know the focal plane crossing locations for the center of the first and sixth SciFi columns in our first bundle, as depicted in Figure 1 by the endpoints of Δx. Additionally, we know the β angle of the first bundle (noted as β_avg above), which gives us the optimal alignment for each fiber in the bundle to their respective electron's path. The β angles for the first and sixth columns will be off by the same magnitude, but with opposite signs.

Figure 2: Sketch showing the path of electrons that pass through the center of the fiber columns near the focal plane. At the back-end of the 2 cm. SciFi a misalignment of around 0.03 mm (with respect the the fiber's axis) results from using an averaged β angle for the bundle support location.

The 5x6 fiber bundle supports were designed with two 5x3 bundle halves offset such that the center of the front face of the middle column in each bundle half would sit on the magnetic focal plane for a β angle of 12.0^o. This angle was selected as a compromise that would allow coverage through the photon energy range of 10 - 5.6 GeV.

As a side note - If required and finances permit, the 17 bundle supports can be easily redesigned for a different β angle. This redesign would take less than an hour of CAD work, with a manufacturing turn-around time in as little as two days. Costs are estimated to be around $5k. A CAD drawing of a new bundle support design, with the current forward and rear bundle half stagger, already exists. This new design incorporates updated locations of the threaded holes for mounting the clamps that keep the bundle straps in place; thereby, permitting a slightly larger range of coverage on the focal plane and a more secure way to hold the fiber straps. The best time to replace/modify the bundle supports, if so desired, would be during fiber replacement. This way the new fibers can be mounted to the new bundle support outside the tagger hall, before ever making it to JLab. A conservative time estimate for changing the TAGM fiber configuration would be approximately two days (16 hours).

Each bundle has a "pivot point" that when placed on the focal plane, provides the optimal y-displacement from the focal plane for each fiber column in that bundle without encroaching too close to the tagger magnet window. If the bundle β angle ≠ 12^o, then the 1^st & 6^th, 2^nd & 5^th, and 3^rd & 4^th fiber column pairs will have the same magnitude offset as one another from the focal plane in Y, but with opposite signs, see Figure 2. This is all provided that when the bundle support is mounted on the parallel railing system the tagger magnet's focal plane passes through the midpoint between the front and rear bundle halves (the so called pivot point).

Figure 3: Sketch showing the "pivot point" that should always sit on the focal plane to provide the optimal fiber alignment. The green line running through the pivot point represent the placement of the focal plane for a bundle support with β = 12^o, while the red line shows the placement for β > 12^o.

As shown in Figure 3 by the red and green lines, if β ≠ 12^o then the front face of the first column of SciFi no longer sits on the focal plane. This offset must be accounted for. The Excel spreadsheet starts by finding the optimal first bundle support crossing angle (β_avg) and places the front center of the first column of SciFi on the X_FP location corresponding to E_γo. Depending on β's departure from 12^o the pivot point will be displaced from the focal plane. The spreadsheet calculates the (Δx, Δy) needed to return the pivot point to the focal plane and keep the first fiber column's longitudinal axis on the E_γo electron's path. The spreadsheet's title for this is "Pivot Point Move" and the description is detailed below.

NOTE: β_avg is used to determine this displacement. While the E_γo electron's β angle differs slightly β_avg and would keep the column's centerline on the proper electron path, this discrepancy is so small that it does not come close to the TAGM machining parts' tolerance. Additionally, the sixth column's β angle has the same magnitude difference from the bundle support's β, but with a different sign. For these reasons β_avg was used.

• Pivot Point Move for β ≠ 12 deg.
• 

  Figure 4: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm² boxes at the end of each 5x3 bundle half are only included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.

• 

  Figure 5: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm² boxes at the 5x3 bundle half are only included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.

• 

  Figure 6: Dimensions of the bundle support as viewed from below. This view shows the location of the bundle support mounting rods at 2.75 in. and 8 in. from the front of the bundle support.

For the next part we assuming the center of the first fiber column's front face is on the focal plane where an E_γo post-bremsstrahlung electron will cross and the bundle support is placed at an angle β_avg

Figure 7: Sketch showing the displacement needed to move the "pivot point" to the focal plane for β ≠ 12^o, while maintaining the first fiber column on the E_γo electron's path.

• Calculate the Y_FP displacement required to place the focal plane back on the bundle support's pivot point, Figure 5
• The angle made by the first fiber column's longitudinal axis and a line from (X₁, 0)_FP to the pivot point is 19.5072^o, see Figure 4
• Use the difference between β_avg and 19.5072^o
• The distance of the center of the first fiber column's front face to the bundle support's pivot point (in the (x, y)_FP plane) is 0.5895 inches
• Note that if β > ~19.5^o (e.g. > 10.86 GeV photons being tagged), then the pivot point will be below the focal plane; therefore, Δy will be positive (bundle shifts towards the magnet, positive Y_FP direction)
• Utilizing the y-displacement value found above and the tangent of β_avg, calculate the associated x-displacement along the focal plane to keep column #1 aligned to the electrons associated with E_γo

Since the post-bremsstrahlung electron's crossing angle changes with energy (e.g. displacement along the focal plane) each bundle support will have a slight "kick" or "tow" from the adjacent bundle support.Simply put, going from upstream [highest E_γ] to downstream [lowest E_γ], subsequent bundle supports will have smaller and smaller β_avg's.

Figure 8: CAD image of two bundle supports (Top View). The blue bundle support was shifted, from the red bundle support's location, along X_FP by 1.2/Sin(β_avg1) [in cm.]. The blue bundle support was then rotated by the tow angle about the left corner. This corner of the second bundle support (blue) is the only part that comes in contact with the first bundle support (red).

Figure 9: Zoomed in version of the CAD image in Figure 8. The image shows how the amount in Y_FP, that the bundle must shift in order to return the pivot point of bundle support #2 to the focal plane after it's β_avg2 has been adjusted for the required "tow", will require a shift in X_FP to return the corner of bundle support #2 as the only contact point with bundle support #1.

Figure 10: Zoomed in version of the CAD image in Figure 8. The dY_2 (ΔY₂) is the amount in Y_FP that the bundle must shift in order to return the pivot point of bundle support #2 to the focal plane after it's β_avg2 has adjusted for the required "tow."

• Next determine the locations of the bundle support mounting rods in focal plane coordinates
• Figure 6 shows the required dimensions for these calculations
• For x locations, include x₁ (for E_γo electrons) in addition to the rods' displacement from the front center of the first fiber column
• For y locations, we assume y₁ (for E_γo electrons) lies on the focal plane y-axis
• The Excel spreadsheet provides calculations for the corners of the upstream bundle halves for each bundle support. As shown in Figure 9 below, it is easiest to determine the forward rod location from the upstream bundle half corner closest to the magnet.
• The rear support rod is 5.25 inches, see Figure 6, from the forward support rod along the bundle support's longitudinal axis.

Figure 9: Image of a lone bundle support. This image shows the center of column #1 on on the focal plane. This is the starting point for bundle support #1, but the equations are also useful in the Excel file when determining the beginning and end energy range covered by each bundle support.

Fitting the Position of Each Bundle Support Mounting Rod to a Straight Line

At this point the exact location for each bundle support rod is known. These locations place the scintillating fibers of each bundle support at their optimal location. Unfortunately these locations deviate slightly from a straight line. To correct this deviation and find a compromise the their location that's within the TAGM required tolerance, we must use ROOT and fit a first order polynomial to the calculated locations. A template C++ file has been written for this purpose. Once the arrays used for the rods' x and y locations are updated for the desired tagging energy spectrum, then the macro can be run or the code can be copy & pasted directly into ROOT's command line. The generated histograms, for the forward and rear rods, can then be fitted to a first order polynomial (pol1) using the Fitting Tool. Several fit equations are already recorded in a fit function repository file. The fits are then utilized back in the Excel file to determine the "Y-Fit" location for each rod X location. This way the parallel rails can be orientated based on these numbers and the three rail components required for the TAGM move can be designed.

• Fitting the calculations to real life
• 

  Figure 10: Histogram of the bundle supports' forward rod position (*) in focal plane coordinates. The red line is a first order polynomial fit (pol1) using the ROOT fitter.

• 

  Figure 11: Histogram of the bundle supports' rear rod position (*) in focal plane coordinates. The red line is a first order polynomial fit (pol1) using the ROOT fitter.

• Copy and paste the locations for the bundle support rods into the the fitting file and run in ROOT.
• Fit the histograms to a straight line (pol1) and input the resulting fit equations into both fit function repository file and the Excel file for calculating bundle support locations
• The Excel file will provide the "Fit-Y" locations that are used in the CAD drawing to design the three unique parallel rail pieces that are needed for each TAGM move.
• Using [http: Beta_Manipulation2.dwg], rotate each bundle support by its β angle then locate the forward rod at the fit position determined by the previous step.
• With all bundle supports in the fit determined location (forward rod), draw a straight line through the center of the forward rods and through the best fit of the rear rods.
• Using these two straight lines realign the parallel rails.
• The Excel file also contains a distance from the parallel rail upstream tapped hole to where the first bundle should start, so as to allow for screw heads to fit.
• Finally use CAD to design the parallel rail support components needed for the move. See below, Figure 12 - 18, for the 6 GeV starting point drawings.

Figure 12: CAD image of a the parallel rails. A 2D image of each bundle support can be seen laying on the top of the parallel rails for reference purposes only.

Figure 13: CAD image of a the upstream top guide plate for the parallel rails. A 2D image of each bundle support can be seen laying on the top of the parallel rails for reference purposes only.

Figure 14: CAD image of a the downstream guide bar for the parallel rails. A 2D image of each bundle support can be seen laying on the top of the parallel rails for reference purposes only.

The CAD drawings (.igs or .stp) for the following components are sent to PROTOLABS for fabrication. Tolerance ±0.005 inches.

Figure 15: CAD image of the upstream guide bar. The upstream guide bar has a notch taken out of its corner for easy identification during installation.

Figure 16: CAD image of the downstream guide bar.

Figure 17: CAD image of the top guide plate.

Figure 18: PROTOLAB quote for the 6 GeV move components.