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An Excel [/nfs/direct/annex/mcintyre/TAGM-4-2021/Bundle-Support-4-2021.xlsx spreadsheet(placeholder)] 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 [https://zeus.phys.uconn.edu/wiki/index.php/Student_Projects_in_Nuclear_Physics drawing(placeholder)] used to design the three parallel rail components needed for each unique tagging energy spectrum starting position of the TAGM and a [https://zeus.phys.uconn.edu/wiki/index.php/Student_Projects_in_Nuclear_Physics drawing(placeholder)] for the current setup (β = 12.5<sup>o</sup> to 11.06<sup>o</sup>). These files are in US standard units (inches) and to scale with a tolerance of ± 0.001 inch.
===<u>A summary of the spreadsheet calculations is a follows:</u>===
[[Image:Bundle_support_displacement.jpg|right|thumb|475px|Figure 16: 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.]]
<ul>
<li style="padding-bottom: 16px;">Select a starting energy for the photon tagging array (highest γ energy to tag)</li>
<li>Using [https://halldweb.jlab.org/wiki/images/b/b6/Counterbounds2017%28b%29.xlsx hodoscope energy bin bounds] interpolate the crossing angle with respect to the focal plane (β<sub>1</sub>) of an electron associated the highest energy to be tagged (E<sub>γ<sub>o</sub></sub>)</li>
<ul>
<li> Interpolate the location on the X<sub>FP</sub> axis at which this electron crosses (X<sub>1</sub>)</li>
<li style="padding-bottom: 16px;"> These electrons will pass through the center of the first column of SciFi fibers</li>
</ul>
<li>Using β<sub>1</sub> 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</li>
<ul>
<li>Add Δx and X<sub>1</sub> to get X<sub>6</sub>, then interpolate the value of β<sub>6</sub></li>
<li>Using the average value of β<sub>1</sub> and β<sub>6</sub> (β<sub>avg.</sub>), recalculate the X<sub>FP</sub> displacement (Δx) from the center of the first to sixth fiber column (X<sub>6</sub>)</li>
<li>Repeat the above two steps until the bundle support angle β (e.g. average between β<sub>1</sub> & β<sub>6</sub>) does not change appreciably</li>
</ul>
</ul>
<p>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 16 by the endpoints of Δx. Additionally, we know the β angle of the first bundle (noted as β<sub>avg</sub> 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.</p>
[[Image:e_path_in_Bundle.png|center|thumb|800px|Figure 17: 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.]]
<p>
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<sup>o</sup>. This angle was selected as a compromise that would allow coverage through the photon energy range of 10 - 5.6 GeV.
</p>
<i><u>As a side note</u> - 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 $2k. A CAD drawing of a new bundle support design already exists, which incorporates updated locations of the threaded holes for mounting the clamps that keep the bundle straps in place. 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).</i>
<p>
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<sup>o</sup>, then the 1<sup><u>st</u></sup> & 6<sup><u>th</u></sup>, 2<sup><u>nd</u></sup> & 5<sup><u>th</u></sup>, and 3<sup><u>rd</u></sup> & 4<sup><u>th</u></sup> fiber column pairs will have the same magnitude offset as one another from the focal plane in Y, but with opposite signs, see Figure 17. 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).
</p>
[[Image:popsicle_stick_fiber_angle_top_view.png|center|thumb|800px|Figure 18: 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<sup>o</sup>, while the red line shows the placement for β > 12<sup>o</sup>.]]
<p>
As shown in Figure 18 by the red and green lines, if β ≠ 12<sup>o</sup> 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<sub>avg</sub>) and places the front center of the first column of SciFi on the X<sub>FP</sub> location corresponding to E<sub>γ<sub>o</sub></sub>. Depending on β's departure from 12<sup>o</sup> 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<sub>γ<sub>o</sub></sub> electron's path. The spreadsheet's title for this is "Pivot Point Move" and the description is detailed below.
</p>
<p>
<i><u>NOTE:</u> β<sub>avg</sub> is used to determine this displacement. While the E<sub>γ<sub>o</sub></sub> electron's β angle differs slightly β<sub>avg</sub> 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<sub>avg</sub> was used.</i>
</p>
<gallery caption="Pivot Point Move for β ≠ 12 deg." widths="450px" heights="450px" class="center">
popsicle_stick_pivot_point.png|Figure 19: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm<sup>2</sup> boxes at the end of each 5x3 bundle half are <i>only</i> included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.
popsicle_stick_fiber_angle8.png|Figure 20: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm<sup>2</sup> boxes at the 5x3 bundle half are <i>only</i> included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.
popsicle_stick_inch_angle.png|Figure 21: 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.
</gallery>
<center><b><u>For the next part we assuming the center of the first fiber column's front face is on the focal plane where an E<sub>γ<sub>o</sub></sub> post-bremsstrahlung electron will cross and the bundle support is placed at an angle β<sub>avg</sub></u></b></center>
[[Image:Bundle_support_FP_shift.jpg|right|thumb|500px|Figure 22: Sketch showing the displacement needed to move the "pivot point" to the focal plane for β ≠ 12<sup>o</sup>, while maintaining the first fiber column on the E<sub>γ<sub>o</sub></sub> electron's path.]]
<ul>
<li>Calculate the Y<sub>FP</sub> displacement required to place the focal plane back on the bundle support's pivot point, Figure 20</li>
<ul>
<li>The angle made by the first fiber column's longitudinal axis and a line from (X<sub>1</sub>, 0)<sub>FP</sub> to the pivot point is 19.5072<sup>o</sup>, see Figure 19</li>
<li>Use the difference between β<sub>avg</sub> and 19.5072<sup>o</sup></li>
<li>The distance of the center of the first fiber column's front face to the bundle support's pivot point (in the (x, y)<sub>FP</sub> plane) is 0.5895 inches</li>
<li style="padding-bottom: 16px;">Note that if β > ~19.5<sup>o</sup> (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<sub>FP</sub> direction)</li>
</ul>
<li style="padding-bottom: 16px;">Utilizing the y-displacement value found above and the tangent of β<sub>avg</sub>, calculate the associated x-displacement along the focal plane to keep column #1 aligned to the electrons associated with E<sub>γ<sub>o</sub></sub></li>
<li>Next determine the locations of the bundle support mounting rods in focal plane coordinates</li>
<ul>
<li>Figure 21 shows the required dimensions for these calculations</li>
<li>For x locations, include x<sub>1</sub> (for E<sub>γ<sub>o</sub></sub> electrons) in addition to the rods' displacement from the front center of the first fiber column</li>
<li>For y locations, we assume y<sub>1</sub> (for E<sub>γ<sub>o</sub></sub> electrons) lies on the focal plane y-axis</li>
</ul>
<b><u><center>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.</center></u>
<i><center>Simply put, going from upstream [highest E<sub>γ</sub>] to downstream [lowest E<sub>γ</sub>], subsequent bundle supports will have smaller and smaller β<sub>avg</sub>'s.</center></i></b>
[[Image:Tow4.png|center|thumb|1000px|Figure 23: CAD image of two bundle supports (Top View). The blue bundle support was shifted, from the red bundle support's location, along X<sub>FP</sub> by 1.2/Sin(β<sub>avg<sub>1</sub></sub>) [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).]]
[[Image:Tow2.png|center|thumb|700px|Figure 24: Zoomed in version of the CAD image in Figure 23. The dY_2 (ΔY<sub>2</sub>) is the amount in Y<sub>FP</sub> that the bundle must shift in order to return the pivot point of bundle support #2 to the focal plane after it's β<sub>avg<sub>2</sub></sub> has adjusted for the required "tow."]]
<li style="padding-bottom: 16px;"></li>
<li style="padding-bottom: 16px;"></li>
</ul>
[[Image:SciFi-on-Popsicle-Stick.png|center|thumb|400px|Figure 12: CAD image of a bundle support (blue) with a mounted 5x6 fiber bundle. The 'S' bend minimizes the amount of fiber in the path of the electrons.]]
===<u>A summary of the spreadsheet calculations is a follows:</u>===
[[Image:Bundle_support_displacement.jpg|right|thumb|475px|Figure 16: 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.]]
<ul>
<li style="padding-bottom: 16px;">Select a starting energy for the photon tagging array (highest γ energy to tag)</li>
<li>Using [https://halldweb.jlab.org/wiki/images/b/b6/Counterbounds2017%28b%29.xlsx hodoscope energy bin bounds] interpolate the crossing angle with respect to the focal plane (β<sub>1</sub>) of an electron associated the highest energy to be tagged (E<sub>γ<sub>o</sub></sub>)</li>
<ul>
<li> Interpolate the location on the X<sub>FP</sub> axis at which this electron crosses (X<sub>1</sub>)</li>
<li style="padding-bottom: 16px;"> These electrons will pass through the center of the first column of SciFi fibers</li>
</ul>
<li>Using β<sub>1</sub> 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</li>
<ul>
<li>Add Δx and X<sub>1</sub> to get X<sub>6</sub>, then interpolate the value of β<sub>6</sub></li>
<li>Using the average value of β<sub>1</sub> and β<sub>6</sub> (β<sub>avg.</sub>), recalculate the X<sub>FP</sub> displacement (Δx) from the center of the first to sixth fiber column (X<sub>6</sub>)</li>
<li>Repeat the above two steps until the bundle support angle β (e.g. average between β<sub>1</sub> & β<sub>6</sub>) does not change appreciably</li>
</ul>
</ul>
<p>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 16 by the endpoints of Δx. Additionally, we know the β angle of the first bundle (noted as β<sub>avg</sub> 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.</p>
[[Image:e_path_in_Bundle.png|center|thumb|800px|Figure 17: 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.]]
<p>
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<sup>o</sup>. This angle was selected as a compromise that would allow coverage through the photon energy range of 10 - 5.6 GeV.
</p>
<i><u>As a side note</u> - 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 $2k. A CAD drawing of a new bundle support design already exists, which incorporates updated locations of the threaded holes for mounting the clamps that keep the bundle straps in place. 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).</i>
<p>
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<sup>o</sup>, then the 1<sup><u>st</u></sup> & 6<sup><u>th</u></sup>, 2<sup><u>nd</u></sup> & 5<sup><u>th</u></sup>, and 3<sup><u>rd</u></sup> & 4<sup><u>th</u></sup> fiber column pairs will have the same magnitude offset as one another from the focal plane in Y, but with opposite signs, see Figure 17. 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).
</p>
[[Image:popsicle_stick_fiber_angle_top_view.png|center|thumb|800px|Figure 18: 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<sup>o</sup>, while the red line shows the placement for β > 12<sup>o</sup>.]]
<p>
As shown in Figure 18 by the red and green lines, if β ≠ 12<sup>o</sup> 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<sub>avg</sub>) and places the front center of the first column of SciFi on the X<sub>FP</sub> location corresponding to E<sub>γ<sub>o</sub></sub>. Depending on β's departure from 12<sup>o</sup> 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<sub>γ<sub>o</sub></sub> electron's path. The spreadsheet's title for this is "Pivot Point Move" and the description is detailed below.
</p>
<p>
<i><u>NOTE:</u> β<sub>avg</sub> is used to determine this displacement. While the E<sub>γ<sub>o</sub></sub> electron's β angle differs slightly β<sub>avg</sub> 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<sub>avg</sub> was used.</i>
</p>
<gallery caption="Pivot Point Move for β ≠ 12 deg." widths="450px" heights="450px" class="center">
popsicle_stick_pivot_point.png|Figure 19: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm<sup>2</sup> boxes at the end of each 5x3 bundle half are <i>only</i> included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.
popsicle_stick_fiber_angle8.png|Figure 20: Dimensions of the pivot point to the front center of the first column of fibers. The three 2 mm<sup>2</sup> boxes at the 5x3 bundle half are <i>only</i> included to represent the width of a fiber column and have no meaning in length in the longitudinal axis direction.
popsicle_stick_inch_angle.png|Figure 21: 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.
</gallery>
<center><b><u>For the next part we assuming the center of the first fiber column's front face is on the focal plane where an E<sub>γ<sub>o</sub></sub> post-bremsstrahlung electron will cross and the bundle support is placed at an angle β<sub>avg</sub></u></b></center>
[[Image:Bundle_support_FP_shift.jpg|right|thumb|500px|Figure 22: Sketch showing the displacement needed to move the "pivot point" to the focal plane for β ≠ 12<sup>o</sup>, while maintaining the first fiber column on the E<sub>γ<sub>o</sub></sub> electron's path.]]
<ul>
<li>Calculate the Y<sub>FP</sub> displacement required to place the focal plane back on the bundle support's pivot point, Figure 20</li>
<ul>
<li>The angle made by the first fiber column's longitudinal axis and a line from (X<sub>1</sub>, 0)<sub>FP</sub> to the pivot point is 19.5072<sup>o</sup>, see Figure 19</li>
<li>Use the difference between β<sub>avg</sub> and 19.5072<sup>o</sup></li>
<li>The distance of the center of the first fiber column's front face to the bundle support's pivot point (in the (x, y)<sub>FP</sub> plane) is 0.5895 inches</li>
<li style="padding-bottom: 16px;">Note that if β > ~19.5<sup>o</sup> (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<sub>FP</sub> direction)</li>
</ul>
<li style="padding-bottom: 16px;">Utilizing the y-displacement value found above and the tangent of β<sub>avg</sub>, calculate the associated x-displacement along the focal plane to keep column #1 aligned to the electrons associated with E<sub>γ<sub>o</sub></sub></li>
<li>Next determine the locations of the bundle support mounting rods in focal plane coordinates</li>
<ul>
<li>Figure 21 shows the required dimensions for these calculations</li>
<li>For x locations, include x<sub>1</sub> (for E<sub>γ<sub>o</sub></sub> electrons) in addition to the rods' displacement from the front center of the first fiber column</li>
<li>For y locations, we assume y<sub>1</sub> (for E<sub>γ<sub>o</sub></sub> electrons) lies on the focal plane y-axis</li>
</ul>
<b><u><center>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.</center></u>
<i><center>Simply put, going from upstream [highest E<sub>γ</sub>] to downstream [lowest E<sub>γ</sub>], subsequent bundle supports will have smaller and smaller β<sub>avg</sub>'s.</center></i></b>
[[Image:Tow4.png|center|thumb|1000px|Figure 23: CAD image of two bundle supports (Top View). The blue bundle support was shifted, from the red bundle support's location, along X<sub>FP</sub> by 1.2/Sin(β<sub>avg<sub>1</sub></sub>) [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).]]
[[Image:Tow2.png|center|thumb|700px|Figure 24: Zoomed in version of the CAD image in Figure 23. The dY_2 (ΔY<sub>2</sub>) is the amount in Y<sub>FP</sub> that the bundle must shift in order to return the pivot point of bundle support #2 to the focal plane after it's β<sub>avg<sub>2</sub></sub> has adjusted for the required "tow."]]
<li style="padding-bottom: 16px;"></li>
<li style="padding-bottom: 16px;"></li>
</ul>
[[Image:SciFi-on-Popsicle-Stick.png|center|thumb|400px|Figure 12: CAD image of a bundle support (blue) with a mounted 5x6 fiber bundle. The 'S' bend minimizes the amount of fiber in the path of the electrons.]]