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===<u>A summary of the spreadsheet calculations is a follows:</u>===
===<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.]]
[[Image:Bundle_support_displacement.jpg|right|thumb|475px|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.]]
<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>
<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 1 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.]]
[[Image:e_path_in_Bundle.png|center|thumb|800px|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.]]
<p>
<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).
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 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).
</p>
</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>.]]
[[Image:popsicle_stick_fiber_angle_top_view.png|center|thumb|800px|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<sup>o</sup>, while the red line shows the placement for β > 12<sup>o</sup>.]]
<p>
<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.
As shown in Figure 3 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>
<gallery caption="Pivot Point Move for β ≠ 12 deg." widths="450px" heights="450px" class="center">
<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_pivot_point.png|Figure 4: 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_fiber_angle8.png|Figure 5: 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.
popsicle_stick_inch_angle.png|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.
</gallery>
</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>
<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|450px|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.]]
[[Image:Bundle_support_FP_shift.jpg|right|thumb|450px|Figure 7: 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>
<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>
<li>Calculate the Y<sub>FP</sub> displacement required to place the focal plane back on the bundle support's pivot point, Figure 5</li>
<ul>
<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>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 4</li>
<li>Use the difference between β<sub>avg</sub> and 19.5072<sup>o</sup></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>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>
<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>
<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:Tow4.png|center|thumb|1000px|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<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:Tow.png|center|thumb|700px|Figure 24: Zoomed in version of the CAD image in Figure 23. The image shows how 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 been adjusted for the required "tow", will require a shift in X<sub>FP</sub> to return the corner of bundle support #2 as the only contact point with bundle support #1.]]
[[Image:Tow.png|center|thumb|700px|Figure 9: Zoomed in version of the CAD image in Figure 8. The image shows how 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 been adjusted for the required "tow", will require a shift in X<sub>FP</sub> to return the corner of bundle support #2 as the only contact point with bundle support #1.]]
[[Image:Tow2.png|center|thumb|700px|Figure 25: 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."]]
[[Image:Tow2.png|center|thumb|700px|Figure 10: Zoomed in version of the CAD image in Figure 8. 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>Next determine the locations of the bundle support mounting rods in focal plane coordinates</li>
<li>Next determine the locations of the bundle support mounting rods in focal plane coordinates</li>
<ul>
<ul>
<li>Figure 21 shows the required dimensions for these calculations</li>
<li>Figure 6 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 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>
<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>
<li>The Excel spreadsheet provides calculations for the corners of the upstream bundle halves for each bundle support. As shown in Figure 26 below, it is easiest to determine the forward rod location from the upstream bundle half corner closest to the magnet.</li>
<li>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.</li>
<li style="padding-bottom: 16px;">The rear support rod is 5.25 inches, see Figure 21, from the forward support rod along the bundle support's longitudinal axis.</li>
<li style="padding-bottom: 16px;">The rear support rod is 5.25 inches, see Figure 6, from the forward support rod along the bundle support's longitudinal axis.</li>
</ul>
</ul>
</ul>
</ul>
[[Image:Rod_position.png|center|thumb|700px|Figure 26: 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.]]
[[Image:Rod_position.png|center|thumb|700px|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==
==Fitting the Position of Each Bundle Support Mounting Rod to a Straight Line==
<gallery caption="Fitting the calculations to real life" widths="450px" heights="450px" class="center">
<gallery caption="Fitting the calculations to real life" widths="450px" heights="450px" class="center">
Fwd_Rod_8GeV.png|Figure 2: 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.
Fwd_Rod_8GeV.png|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.
Rear_Rod_8GeV.png|Figure 3: 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.
Rear_Rod_8GeV.png|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.
</gallery>
</gallery>
</ul>
</ul>
<ul>
<ul>
<li style="padding-bottom: 16px;">Finally use CAD to design the parallel rail support components needed for the move. See below, Figure 27 - 30, for the 6 GeV starting point drawings.</li>
<li style="padding-bottom: 16px;">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.</li>
</ul>
</ul>
</ul>
</ul>
[[Image:parallel_railing.png|center|thumb|1200px|Figure 27: 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.]]
[[Image:parallel_railing.png|center|thumb|1200px|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.]]
[[Image:Upstream_parallel_rail.png|center|thumb|1200px|Figure 28: 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.]]
[[Image:Upstream_parallel_rail.png|center|thumb|1200px|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.]]
[[Image:Downstream_parallel_rail.png|center|thumb|1200px|Figure 29: 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.]]
[[Image:Downstream_parallel_rail.png|center|thumb|1200px|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.]]
<center><b><u>The CAD drawings (.igs or .stp) for the following components are sent to [https://www.protolabs.com/ PROTOLABS] for fabrication.</u></b> Tolerance ±0.005 inches.</center>
<center><b><u>The CAD drawings (.igs or .stp) for the following components are sent to [https://www.protolabs.com/ PROTOLABS] for fabrication.</u></b> Tolerance ±0.005 inches.</center>
[[Image:Upstream_Bar.png|center|thumb|450px|Figure 30: CAD image of the upstream guide bar. The upstream guide bar has a notch taken out of its corner for easy identification during installation.]]
[[Image:Upstream_Bar.png|center|thumb|450px|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.]]
[[Image:Downstream_Bar.png|center|thumb|450px|Figure 31: CAD image of the downstream guide bar.]]
[[Image:Downstream_Bar.png|center|thumb|450px|Figure 16: CAD image of the downstream guide bar.]]
[[Image:Top_Plate.png|center|thumb|750px|Figure 32: CAD image of the top guide plate.]]
[[Image:Top_Plate.png|center|thumb|750px|Figure 17: CAD image of the top guide plate.]]
[[Image:quote.png|center|thumb|750px|Figure 33: PROTOLAB [[media:Quote 7048-123.pdf|quote]] for the 6 GeV move components.]]
[[Image:quote.png|center|thumb|750px|Figure 18: PROTOLAB [[media:Quote 7048-123.pdf|quote]] for the 6 GeV move components.]]