<html><head><title></title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style2, li.Style2 /*Heading 1*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 13.26315789473684pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style3, li.Style3 /*Heading 2*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: bold; font-style: italic; vertical-align: baseline; } p.Style4, li.Style4 /*Heading 3*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style5, li.Style5 /*Paragraph*/ { margin-left: 0pt; margin-right: 0pt; text-indent: 21pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style6, li.Style6 /*List*/ { margin-left: 14.25pt; margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-1121" style="position: absolute; top: 24.75pt; left: 18pt; width: 246.5pt; "><a name="" /><p class="Style1"><span style="font-size: 18pt; "><b><span style="color: #f00; ">THIS IS WRONG DON'T USE</span></b></span></p></div><div id="mc-region-824" style="position: absolute; top: 83.25pt; left: 6pt; width: 426.5pt; "><a name="" /><p class="Style1"><span style="font-size: 13.26315789473684pt; "><b>A simple, accurate formula for evaluating track - hit association using pads over strip detector elements. The formula directly handles the fake hits associated with this geometry.</b></span></p></div><div id="mc-region-827" style="position: absolute; top: 152.25pt; left: 6pt; width: 57.5pt; "><a name="" /><p class="Style1">H. Wieman</p><p class="Style1">23-Jan-2007</p></div><div id="mc-region-829" style="position: absolute; top: 188.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1"><span style="font-size: 10.266666666666667pt; "><b>Abstract </b></span></p><p class="Style1">This note contains a derivation of a formula for the probability of associating hits with tracks for crossed detector elements that generate ambiguous hits. The derivation is verified with a Monte Carlo simulation and some example detector configurations are evaluated as a function of hit density. </p></div><div id="mc-region-831" style="position: absolute; top: 258.75pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Introduction</b></span></p><p class="Style1">A detector unit composed of pads overlaid on strips as shown in fig. 1 can have ambiguous hits when there is more than one hit in the unit. These extra hits dilute the true hit population reducing the probability of correctly associating tracks with the correct hit, but the addition of pads improve the position resolution which improves hit to track association. This analysis includes these competing features and provides a simple quantitative measure of the hit to track association. Hit to track association is done using the minimum weighted chi-square selection method.</p><p class="Style1"></p><p class="Style1"><span style="font-size: 12pt; "><b>Items in order of presentation:</b></span></p><p class="Style1"></p><p class="Style1">1. Formula for probability of correct hit to track association</p><p class="Style1"></p><p class="Style1">2. Evaluation of three different detector unit configurations using the formula</p><p class="Style1"></p><p class="Style1">3. Derivation of the formula</p><p class="Style1"></p><p class="Style1">4. Confirmation of formula with Monte Carlo test</p><p class="Style1"></p><p class="Style1">5. Conclusions </p></div><div id="mc-region-1116" style="position: absolute; top: 516pt; left: 6pt; width: 412.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0003_22561296.PNG" id="generatedImage1116" path=".\HitFindingPadVsStrip_images/IMG0003_22561296.PNG" style="width: 412.5pt;height: 174pt;"></img></div><div id="mc-region-834" style="position: absolute; top: 710.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">Fig. 1 The diagram shows a detector unit with pads overlaid on strips. The example shown has three real hits shown as x's. This results in three true detector determined hits shown as solid circles and 6 fake hits shown as open circles. There is a ambiguity between the true hits and the fake hits and all hits are considered in matching hits to tracks. </p></div><div id="mc-region-842" style="position: absolute; top: 780.75pt; left: 6pt; width: 65pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>1. Formula</b></span></p></div><div id="mc-region-839" style="position: absolute; top: 818.25pt; left: 6pt; width: 260pt; "><a name="" /><p class="Style1">The probability of correctly associating a hit with a track is:</p></div><div id="mc-region-836" style="position: absolute; top: 837.75pt; left: 6pt; width: 297.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0004_22561296.PNG" id="generatedImage836" path=".\HitFindingPadVsStrip_images/IMG0004_22561296.PNG" style="width: 297.75pt;height: 72.75pt;"></img></div><div id="mc-region-1113" style="position: absolute; top: 890.25pt; left: 372pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 1</p></div><div id="mc-region-841" style="position: absolute; top: 926.25pt; left: 6pt; width: 30.5pt; "><a name="" /><p class="Style1">where</p></div><div id="mc-region-847" style="position: absolute; top: 962.25pt; left: 30pt; width: 7.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0005_22561296.PNG" id="generatedImage847" path=".\HitFindingPadVsStrip_images/IMG0005_22561296.PNG" style="width: 7.5pt;height: 13.5pt;"></img></div><div id="mc-region-846" style="position: absolute; top: 962.25pt; left: 84pt; width: 183.5pt; "><a name="" /><p class="Style1">is the true hit density on the detector unit</p></div><div id="mc-region-848" style="position: absolute; top: 993pt; left: 30pt; width: 13.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0006_22561312.PNG" id="generatedImage848" path=".\HitFindingPadVsStrip_images/IMG0006_22561312.PNG" style="width: 13.5pt;height: 16.5pt;"></img></div><div id="mc-region-849" style="position: absolute; top: 992.25pt; left: 90pt; width: 317pt; "><a name="" /><p class="Style1">is the area of the crossed detector unit. Working from fig. 1 this area is:</p></div><div id="mc-region-850" style="position: absolute; top: 1017pt; left: 120pt; width: 28.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0007_22561312.PNG" id="generatedImage850" path=".\HitFindingPadVsStrip_images/IMG0007_22561312.PNG" style="width: 28.5pt;height: 12.75pt;"></img></div><div id="mc-region-851" style="position: absolute; top: 1047pt; left: 30pt; width: 13.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0008_22561312.PNG" id="generatedImage851" path=".\HitFindingPadVsStrip_images/IMG0008_22561312.PNG" style="width: 13.5pt;height: 16.5pt;"></img></div><div id="mc-region-853" style="position: absolute; top: 1046.25pt; left: 90pt; width: 250.25pt; "><a name="" /><p class="Style1">is the area of the detector pad. From fig. 1 this area is: </p></div><div id="mc-region-888" style="position: absolute; top: 1077pt; left: 120pt; width: 29.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0009_22561312.PNG" id="generatedImage888" path=".\HitFindingPadVsStrip_images/IMG0009_22561312.PNG" style="width: 29.25pt;height: 12.75pt;"></img></div><div id="mc-region-855" style="position: absolute; top: 1100.25pt; left: 30pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0010_22561312.PNG" id="generatedImage855" path=".\HitFindingPadVsStrip_images/IMG0010_22561312.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-857" style="position: absolute; top: 1100.25pt; left: 90pt; width: 354.5pt; "><a name="" /><p class="Style1">is the combined uncertainty in the x direction from track projection error and detector uncertainty. In the examples we work with the track projection error and the detector error from the length of the pads are comparable so both should be added in quadrature.</p></div><div id="mc-region-859" style="position: absolute; top: 1178.25pt; left: 36pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0011_22561312.PNG" id="generatedImage859" path=".\HitFindingPadVsStrip_images/IMG0011_22561312.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-861" style="position: absolute; top: 1178.25pt; left: 90pt; width: 354.5pt; "><a name="" /><p class="Style1">is the combined uncertainty in the y direction from track projection error and detector uncertainty. In the examples we work with the track projection error dominates and the detector error from the width of the strips can be neglected.</p></div><div id="mc-region-862" style="position: absolute; top: 1243.5pt; left: 42pt; width: 92.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0012_22561312.PNG" id="generatedImage862" path=".\HitFindingPadVsStrip_images/IMG0012_22561312.PNG" style="width: 92.25pt;height: 34.5pt;"></img></div><div id="mc-region-864" style="position: absolute; top: 1256.25pt; left: 168pt; width: 104pt; "><a name="" /><p class="Style1">is the Poisson function</p></div><div id="mc-region-873" style="position: absolute; top: 1304.25pt; left: 36pt; width: 404.75pt; "><a name="" /><p class="Style1">Note, in practice only a few terms of the sum are required as the series quickly dies. There is probably a simple expression for the infinite series, but Math Cad didn't find it. If someone knows what it is let me know.</p></div><div id="mc-region-866" style="position: absolute; top: 1352.25pt; left: 12pt; width: 218pt; "><a name="" /><p class="Style1">Some assumptions that this formula is based on:</p></div><div id="mc-region-871" style="position: absolute; top: 1382.25pt; left: 42pt; width: 402.5pt; "><a name="" /><p class="Style1">Every track has a true hit on the detector. If this is not the case then the above probability gets multiplied by the detector efficiency for hit detection. The detected hit density, however, still goes into the formula argument.</p><p class="Style1"></p><p class="Style1">An infinite search window is used.</p><p class="Style1"></p><p class="Style1">In doing the association hits are not removed from the pool when they are associated with a track. As <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsqMC.htm">discussed previously</a></span></u></b> this is the best way to do the association, particularly when track densities are high.</p><p class="Style1"></p><p class="Style1">There is no hit noise in the detector. If there is hit noise then the formula should be altered to include this effect. For example noise that triggers a pad but for which there are no hits on a strip, that would not be recorded as a hit in the detector unit.</p><p class="Style1"></p><p class="Style1">Hit to track association is made by picking pairs with the minimum value for the function:</p></div><div id="mc-region-876" style="position: absolute; top: 1616.25pt; left: 258pt; width: 179pt; "><a name="" /><p class="Style1">weighted chi-square function used for hit association</p></div><div id="mc-region-874" style="position: absolute; top: 1606.5pt; left: 48pt; width: 186pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0013_22561312.PNG" id="generatedImage874" path=".\HitFindingPadVsStrip_images/IMG0013_22561312.PNG" style="width: 186pt;height: 48pt;"></img></div><div id="mc-region-877" style="position: absolute; top: 1677pt; left: 72pt; width: 24.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0014_22561328.PNG" id="generatedImage877" path=".\HitFindingPadVsStrip_images/IMG0014_22561328.PNG" style="width: 24.75pt;height: 12.75pt;"></img></div><div id="mc-region-887" style="position: absolute; top: 1701pt; left: 96pt; width: 27.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0015_22561328.PNG" id="generatedImage887" path=".\HitFindingPadVsStrip_images/IMG0015_22561328.PNG" style="width: 27.75pt;height: 16.5pt;"></img></div><div id="mc-region-881" style="position: absolute; top: 1700.25pt; left: 186pt; width: 201.5pt; "><a name="" /><p class="Style1">is the projected track location on the detector</p></div><div id="mc-region-886" style="position: absolute; top: 1743pt; left: 96pt; width: 32.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0016_22561328.PNG" id="generatedImage886" path=".\HitFindingPadVsStrip_images/IMG0016_22561328.PNG" style="width: 32.25pt;height: 16.5pt;"></img></div><div id="mc-region-885" style="position: absolute; top: 1742.25pt; left: 186pt; width: 198.5pt; "><a name="" /><p class="Style1">is the hit location determined by the detector</p></div><div id="mc-region-890" style="position: absolute; top: 1794.75pt; left: 6pt; width: 336.5pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>2. Evaluation of three different detector unit configurations</b></span></p></div><div id="mc-region-892" style="position: absolute; top: 1832.25pt; left: 6pt; width: 403.25pt; "><a name="" /><p class="Style1">Three different configurations are evaluated. The first two are like the illustration in fig. 1 with different geometries and the third one is with <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsqMC.htm">single strips, no pads as was evaluated previously</a></span></u></b>.</p></div><div id="mc-region-894" style="position: absolute; top: 1868.25pt; left: 6pt; width: 395.75pt; "><a name="" /><p class="Style1">The <b>first geometry</b> of pads and strips is what I believe Gerrit van Nieuwenhuizen has been proposing, namely 20 pads over 20 strips.</p></div><div id="mc-region-1107" style="position: absolute; top: 1917pt; left: 18pt; width: 54.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0017_22561328.PNG" id="generatedImage1107" path=".\HitFindingPadVsStrip_images/IMG0017_22561328.PNG" style="width: 54.75pt;height: 16.5pt;"></img></div><div id="mc-region-1111" style="position: absolute; top: 1916.25pt; left: 132pt; width: 186.5pt; "><a name="" /><p class="Style1">Width of pad, vertical direction, y direction</p></div><div id="mc-region-1106" style="position: absolute; top: 1947pt; left: 18pt; width: 50.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0018_22561328.PNG" id="generatedImage1106" path=".\HitFindingPadVsStrip_images/IMG0018_22561328.PNG" style="width: 50.25pt;height: 16.5pt;"></img></div><div id="mc-region-1110" style="position: absolute; top: 1946.25pt; left: 132pt; width: 202.25pt; "><a name="" /><p class="Style1">Length of pad, horizontal direction, x direction</p></div><div id="mc-region-1105" style="position: absolute; top: 1977pt; left: 18pt; width: 54pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0019_22561328.PNG" id="generatedImage1105" path=".\HitFindingPadVsStrip_images/IMG0019_22561328.PNG" style="width: 54pt;height: 16.5pt;"></img></div><div id="mc-region-1109" style="position: absolute; top: 1976.25pt; left: 132pt; width: 158pt; "><a name="" /><p class="Style1">Width of strip, vertical or y direction</p></div><div id="mc-region-1104" style="position: absolute; top: 2001pt; left: 18pt; width: 47.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0020_22561328.PNG" id="generatedImage1104" path=".\HitFindingPadVsStrip_images/IMG0020_22561328.PNG" style="width: 47.25pt;height: 16.5pt;"></img></div><div id="mc-region-1108" style="position: absolute; top: 2000.25pt; left: 132pt; width: 173.75pt; "><a name="" /><p class="Style1">Length of strip, horizontal or x direction</p></div><div id="mc-region-908" style="position: absolute; top: 2043pt; left: 18pt; width: 54pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0021_22561343.PNG" id="generatedImage908" path=".\HitFindingPadVsStrip_images/IMG0021_22561343.PNG" style="width: 54pt;height: 16.5pt;"></img></div><div id="mc-region-909" style="position: absolute; top: 2042.25pt; left: 132pt; width: 69.5pt; "><a name="" /><p class="Style1">area of the pad</p></div><div id="mc-region-913" style="position: absolute; top: 2084.25pt; left: 168pt; width: 263.75pt; "><a name="" /><p class="Style1">detector resolution added in quadrature with track projection error</p></div><div id="mc-region-911" style="position: absolute; top: 2073pt; left: 18pt; width: 112.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0022_22561343.PNG" id="generatedImage911" path=".\HitFindingPadVsStrip_images/IMG0022_22561343.PNG" style="width: 112.5pt;height: 40.5pt;"></img></div><div id="mc-region-914" style="position: absolute; top: 2144.25pt; left: 24pt; width: 49.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0023_22561343.PNG" id="generatedImage914" path=".\HitFindingPadVsStrip_images/IMG0023_22561343.PNG" style="width: 49.5pt;height: 17.25pt;"></img></div><div id="mc-region-917" style="position: absolute; top: 2144.25pt; left: 162pt; width: 282.5pt; "><a name="" /><p class="Style1">track projection error. The detector error in this direction is insignificant by comparison:</p></div><div id="mc-region-919" style="position: absolute; top: 2164.5pt; left: 324pt; width: 19.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0024_22561343.PNG" id="generatedImage919" path=".\HitFindingPadVsStrip_images/IMG0024_22561343.PNG" style="width: 19.5pt;height: 33pt;"></img></div><div id="mc-region-920" style="position: absolute; top: 2229pt; left: 24pt; width: 53.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0025_22561343.PNG" id="generatedImage920" path=".\HitFindingPadVsStrip_images/IMG0025_22561343.PNG" style="width: 53.25pt;height: 16.5pt;"></img></div><div id="mc-region-922" style="position: absolute; top: 2228.25pt; left: 156pt; width: 146.75pt; "><a name="" /><p class="Style1">Area of the crossed detector unit</p></div><div id="mc-region-927" style="position: absolute; top: 2270.25pt; left: 24pt; width: 191.75pt; "><a name="" /><p class="Style1">In this analysis we use 3 terms in the sum:</p></div><div id="mc-region-954" style="position: absolute; top: 2313.75pt; left: 18pt; width: 300pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0026_22561343.PNG" id="generatedImage954" path=".\HitFindingPadVsStrip_images/IMG0026_22561343.PNG" style="width: 300pt;height: 72.75pt;"></img></div><div id="mc-region-959" style="position: absolute; top: 2438.25pt; left: 330pt; width: 60.5pt; "><a name="" /><p class="Style1">Gerrit design</p></div><div id="mc-region-953" style="position: absolute; top: 2427pt; left: 18pt; width: 261pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0027_22561343.PNG" id="generatedImage953" path=".\HitFindingPadVsStrip_images/IMG0027_22561343.PNG" style="width: 261pt;height: 40.5pt;"></img></div><div id="mc-region-931" style="position: absolute; top: 2504.25pt; left: 6pt; width: 228.5pt; "><a name="" /><p class="Style1">The <b>second geometry</b> was proposed by Ivan Kotov</p></div><div id="mc-region-933" style="position: absolute; top: 2534.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">It has the same electronic density as the Gerrit's design. The difference is that the detector unit size is half the size by making the pad length twice as long and the pad width half the value. This gives a lower probability of ambiguous hits from multiple hits in a unit at the expense of poorer resolution in the x direction. </p></div><div id="mc-region-936" style="position: absolute; top: 2600.25pt; left: 36pt; width: 382.25pt; "><a name="" /><p class="Style1">So with this geometry expressed in terms of the original dimensions of pads and strips:</p></div><div id="mc-region-937" style="position: absolute; top: 2626.5pt; left: 30pt; width: 56.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0028_22561343.PNG" id="generatedImage937" path=".\HitFindingPadVsStrip_images/IMG0028_22561343.PNG" style="width: 56.25pt;height: 31.5pt;"></img></div><div id="mc-region-952" style="position: absolute; top: 2636.25pt; left: 204pt; width: 80.75pt; "><a name="" /><p class="Style1">detector unit area</p></div><div id="mc-region-950" style="position: absolute; top: 2673pt; left: 30pt; width: 114.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0029_22561359.PNG" id="generatedImage950" path=".\HitFindingPadVsStrip_images/IMG0029_22561359.PNG" style="width: 114.75pt;height: 40.5pt;"></img></div><div id="mc-region-951" style="position: absolute; top: 2690.25pt; left: 204pt; width: 206.75pt; "><a name="" /><p class="Style1">combined detector and projection error in the x direction</p></div><div id="mc-region-944" style="position: absolute; top: 2744.25pt; left: 30pt; width: 194.75pt; "><a name="" /><p class="Style1">The rest of the parameters remain the same</p></div><div id="mc-region-962" style="position: absolute; top: 2792.25pt; left: 348pt; width: 53pt; "><a name="" /><p class="Style1">Ivan design</p></div><div id="mc-region-943" style="position: absolute; top: 2781pt; left: 30pt; width: 264pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0030_22561359.PNG" id="generatedImage943" path=".\HitFindingPadVsStrip_images/IMG0030_22561359.PNG" style="width: 264pt;height: 40.5pt;"></img></div><div id="mc-region-956" style="position: absolute; top: 2846.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">The <b>third geometry</b> is just a single strip in which case we use the same function that was used in a <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsqMC.htm">prior note</a></span></u></b>. Multiple hits on a strip do not change anything since the detector records the same information whether there are one, two or more hits on the strip.</p></div><div id="mc-region-965" style="position: absolute; top: 2900.25pt; left: 54pt; width: 106.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0031_22561359.PNG" id="generatedImage965" path=".\HitFindingPadVsStrip_images/IMG0031_22561359.PNG" style="width: 106.5pt;height: 32.25pt;"></img></div><div id="mc-region-966" style="position: absolute; top: 2978.25pt; left: 42pt; width: 138pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0032_22561390.PNG" id="generatedImage966" path=".\HitFindingPadVsStrip_images/IMG0032_22561390.PNG" style="width: 138pt;height: 48pt;"></img></div><div id="mc-region-1103" style="position: absolute; top: 2990.25pt; left: 288pt; width: 52.25pt; "><a name="" /><p class="Style1">single strip</p></div><div id="mc-region-975" style="position: absolute; top: 3078pt; left: 6pt; width: 331.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0034_22561390.PNG" id="generatedImage975" path=".\HitFindingPadVsStrip_images/IMG0034_22561390.PNG" style="width: 331.5pt;height: 360pt;"></img></div><div id="mc-region-1102" style="position: absolute; top: 3470.25pt; left: 6pt; width: 432.5pt; "><a name="" /><p class="Style1">Fig. 2 A comparison of detector configurations. Both designs with pads over strips are superior to strips only in the region of expected maximum hit densities, 0.7 to 1.2 hits/cm<sup>2</sup> </p></div><div id="mc-region-979" style="position: absolute; top: 3515.25pt; left: 12pt; width: 410.75pt; "><a name="" /><p class="Style1">At 1.2 hits/cm<span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sup>2</sup></span></span>, the maximum predicted U+U hit density the two pad designs give virtually the same correct association probability:</p></div><div id="mc-region-981" style="position: absolute; top: 3554.25pt; left: 36pt; width: 78.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0035_22561390.PNG" id="generatedImage981" path=".\HitFindingPadVsStrip_images/IMG0035_22561390.PNG" style="width: 78.75pt;height: 33.75pt;"></img></div><div id="mc-region-983" style="position: absolute; top: 3554.25pt; left: 174pt; width: 76.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0036_22561390.PNG" id="generatedImage983" path=".\HitFindingPadVsStrip_images/IMG0036_22561390.PNG" style="width: 76.5pt;height: 33.75pt;"></img></div><div id="mc-region-985" style="position: absolute; top: 3614.25pt; left: 12pt; width: 410pt; "><a name="" /><p class="Style1">So, one would choose the Gerrit design with a larger detector unit size, but shorter pad length because it gives a better detector position resolution.</p></div><div id="mc-region-986" style="position: absolute; top: 3672.75pt; left: 12pt; width: 155pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>3 Derivation of the formula</b></span></p></div><div id="mc-region-988" style="position: absolute; top: 3711.75pt; left: 6pt; width: 297.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0037_22561390.PNG" id="generatedImage988" path=".\HitFindingPadVsStrip_images/IMG0037_22561390.PNG" style="width: 297.75pt;height: 72.75pt;"></img></div><div id="mc-region-997" style="position: absolute; top: 3758.25pt; left: 360pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 1</p></div><div id="mc-region-990" style="position: absolute; top: 3800.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">The derivation of this formula (eq. 1) makes repeated use of the result (eq. 2) found in a <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsq.htm">previous note</a></span></u></b> which gives the probability for a correctly associating hits with track projections with a single detector element: </p></div><div id="mc-region-1001" style="position: absolute; top: 3854.25pt; left: 72pt; width: 70.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0038_22561406.PNG" id="generatedImage1001" path=".\HitFindingPadVsStrip_images/IMG0038_22561406.PNG" style="width: 70.5pt;height: 32.25pt;"></img></div><div id="mc-region-1000" style="position: absolute; top: 3860.25pt; left: 192pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 2</p></div><div id="mc-region-995" style="position: absolute; top: 3908.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">The first term of eq. 1 is just eq. 2 weighted by the probability of a single hit in the detector unit, given by the Poisson function, P(1,A<span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sub>u</sub></span></span>*<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">r</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">)</span></span>. The <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">s</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> s are the combined detector element position uncertainties and the track projection uncertainty. The hit density in the case of the first term is just the raw hit density.</span></span></p><p class="Style1"></p><p class="Style1"><span style="font-weight: normal; font-style: normal;"><span style="font-size: 10pt; font-family: 'Arial', sans-serif;">The terms in the sum handle cases with there are 2, 3, 4 ... real hits in the detector unit. Again we use eq. 2, but now the density of hits that can be confused with the real hit is given by the density of fake hits which is (n-1)/A</span></span><sub><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">p</span></span></sub><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">. Where n is the number of true hits in the detector unit and A</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sub>p</sub></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> is the area of the pad. In this case we have to be more careful about counting and not include the true hit since we are dealing with small numbers. That is why we us n-1. The Poisson function, P(n,A</span></span><span style="font-weight: normal; font-style: normal;"><sub><span style="font-size: 10pt; font-family: 'Arial', sans-serif;">u</span></sub></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">*</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">r</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">),</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> gives the probability for finding n real hits in the detector unit. The denominator of eq. 1 is the total probability of having a detector unit with 1 or more hits.</span></span></p></div><div id="mc-region-1013" style="position: absolute; top: 4086.75pt; left: 12pt; width: 266pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>4 Confirmation of eq. 1 with a Monte Carlo test</b></span></p></div><div id="mc-region-1018" style="position: absolute; top: 4124.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">The following figure shows a comparison of Monte Carlo with the terms in the sum of eq. 1. This is done for n = 2 through 20. The Poisson weighting factor has not been included. This part tests the new part of the analysis which is the handling of fake hits introduced by the crossing of pads with strips. The Gerrit's design geometry with 20 pads on 20 strips was used for this demonstration. The Monte Carlo was performed by distributing n tracks uniformly over the detector unit. The true hits and fake hits from crossing ambiguity were assigned to pad and strip centers as they would be in a real device. The original track locations were dithered with a norm distribution appropriate for tracking resolution. The <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">s</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">s used in the formula (eq. 3) and the chi-square test include both the tracking uncertainty and the width of detector elements divided by root 12.</span></span> </p></div><div id="mc-region-1030" style="position: absolute; top: 4268.25pt; left: 6pt; width: 212pt; "><a name="" /><p class="Style1">Parameters in the comparison, units are in mm:</p></div><div id="mc-region-1059" style="position: absolute; top: 4293pt; left: 6pt; width: 33.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0039_22561406.PNG" id="generatedImage1059" path=".\HitFindingPadVsStrip_images/IMG0039_22561406.PNG" style="width: 33.75pt;height: 12.75pt;"></img></div><div id="mc-region-1070" style="position: absolute; top: 4292.25pt; left: 78pt; width: 53pt; "><a name="" /><p class="Style1">strip length</p></div><div id="mc-region-1060" style="position: absolute; top: 4317pt; left: 6pt; width: 40.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0040_22561406.PNG" id="generatedImage1060" path=".\HitFindingPadVsStrip_images/IMG0040_22561406.PNG" style="width: 40.5pt;height: 12.75pt;"></img></div><div id="mc-region-1069" style="position: absolute; top: 4316.25pt; left: 78pt; width: 49.25pt; "><a name="" /><p class="Style1">strip width</p></div><div id="mc-region-1061" style="position: absolute; top: 4341pt; left: 6pt; width: 36.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0041_22561406.PNG" id="generatedImage1061" path=".\HitFindingPadVsStrip_images/IMG0041_22561406.PNG" style="width: 36.75pt;height: 12.75pt;"></img></div><div id="mc-region-1068" style="position: absolute; top: 4340.25pt; left: 78pt; width: 50pt; "><a name="" /><p class="Style1">pad length</p></div><div id="mc-region-1062" style="position: absolute; top: 4371pt; left: 6pt; width: 41.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0042_22561406.PNG" id="generatedImage1062" path=".\HitFindingPadVsStrip_images/IMG0042_22561406.PNG" style="width: 41.25pt;height: 12.75pt;"></img></div><div id="mc-region-1067" style="position: absolute; top: 4370.25pt; left: 78pt; width: 46.25pt; "><a name="" /><p class="Style1">pad width</p></div><div id="mc-region-1063" style="position: absolute; top: 4400.25pt; left: 6pt; width: 33.75pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0043_22561406.PNG" id="generatedImage1063" path=".\HitFindingPadVsStrip_images/IMG0043_22561406.PNG" style="width: 33.75pt;height: 13.5pt;"></img></div><div id="mc-region-1066" style="position: absolute; top: 4400.25pt; left: 78pt; width: 353pt; "><a name="" /><p class="Style1">track projection uncertainty x direction, used to dither Monte Carlo track location</p></div><div id="mc-region-1064" style="position: absolute; top: 4430.25pt; left: 6pt; width: 35.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0044_22561406.PNG" id="generatedImage1064" path=".\HitFindingPadVsStrip_images/IMG0044_22561406.PNG" style="width: 35.25pt;height: 13.5pt;"></img></div><div id="mc-region-1065" style="position: absolute; top: 4430.25pt; left: 78pt; width: 353pt; "><a name="" /><p class="Style1">track projection uncertainty y direction, used to dither Monte Carlo track location</p></div><div id="mc-region-1071" style="position: absolute; top: 4470.75pt; left: 6pt; width: 111pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0045_22561421.PNG" id="generatedImage1071" path=".\HitFindingPadVsStrip_images/IMG0045_22561421.PNG" style="width: 111pt;height: 36.75pt;"></img></div><div id="mc-region-1088" style="position: absolute; top: 4484.25pt; left: 186pt; width: 131.75pt; "><a name="" /><p class="Style1"><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> used in eq. 3 and chi-sq test</span></span></p></div><div id="mc-region-1076" style="position: absolute; top: 4544.25pt; left: 186pt; width: 128.75pt; "><a name="" /><p class="Style1"><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">used in eq. 3 and chi-sq test</span></span></p></div><div id="mc-region-1072" style="position: absolute; top: 4536.75pt; left: 6pt; width: 114pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0046_22561421.PNG" id="generatedImage1072" path=".\HitFindingPadVsStrip_images/IMG0046_22561421.PNG" style="width: 114pt;height: 36.75pt;"></img></div><div id="mc-region-1090" style="position: absolute; top: 4610.25pt; left: 6pt; width: 173.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0047_22561421.PNG" id="generatedImage1090" path=".\HitFindingPadVsStrip_images/IMG0047_22561421.PNG" style="width: 173.25pt;height: 42.75pt;"></img></div><div id="mc-region-1097" style="position: absolute; top: 4616.25pt; left: 228pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 3</p></div><div id="mc-region-1095" style="position: absolute; top: 4616.25pt; left: 282pt; width: 150.5pt; "><a name="" /><p class="Style1">formula from sum in eq. 1 that is tested against Monte Carlo</p></div><div id="mc-region-1081" style="position: absolute; top: 4678.5pt; left: 6pt; width: 188.25pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0048_22561421.PNG" id="generatedImage1081" path=".\HitFindingPadVsStrip_images/IMG0048_22561421.PNG" style="width: 188.25pt;height: 44.25pt;"></img></div><div id="mc-region-1082" style="position: absolute; top: 4694.25pt; left: 240pt; width: 179pt; "><a name="" /><p class="Style1">weighted chi-square function used for hit association</p></div><div id="mc-region-1009" style="position: absolute; top: 4746pt; left: 30pt; width: 331.5pt; "><a name="" /><img border="0" src=".\HitFindingPadVsStrip_images/IMG0051_22561421.PNG" id="generatedImage1009" path=".\HitFindingPadVsStrip_images/IMG0051_22561421.PNG" style="width: 331.5pt;height: 315pt;"></img></div><div id="mc-region-1011" style="position: absolute; top: 5084.25pt; left: 18pt; width: 416.75pt; "><a name="" /><p class="Style1">Fig 3 Comparison of terms in the sum of eq. 1 with a Monte Carlo test. The Poisson weighting has not been included here. Clearly, the formula gives a good approximation.</p></div><div id="mc-region-1117" style="position: absolute; top: 5148.75pt; left: 6pt; width: 80.75pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>5 Conclusion</b></span></p></div><div id="mc-region-1119" style="position: absolute; top: 5180.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">The derived formula accurately represents the probability of correct hit to track association using realistic geometries and hit densities and can be used as a quick tool to evaluate different pad over strip detector configurations.</p><p class="Style1"></p><p class="Style1"></p><p class="Style1">Pads over strips are clearly preferred over strips alone, both because of better hit to track association and because of better detector position resolution.</p><p class="Style1"></p><p class="Style1"></p><p class="Style1">The two pad/strip configurations tested have nearly the same ability to correctly assign hits to tracks for the hit densities expect at 12 cm radius in STAR, but the one with shorter pads is desirable because of better position resolution.</p></div><div></div></body></html>