<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-indent: 0pt; text-align: left; list-style-type: none; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; color: #00f; } </style></head><body><div id="mc-region-2562" style="position: absolute; top: -0.75pt; left: 12pt; width: 413pt; "><a name="" /><p class="Style1"><span style="font-size: 13.26315789473684pt; ">Space Charge Distortion in the STAR TPC, An Approximate Formula</span></p></div><div id="mc-region-2566" style="position: absolute; top: 20.25pt; left: 30pt; width: 53.75pt; "><a name="" /><p class="Style1">H. Wieman</p><p class="Style1">10/9/2006</p></div><div id="mc-region-2868" style="position: absolute; top: 68.25pt; left: 6pt; width: 424.25pt; "><a name="" /><p class="Style1">Abstract:</p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">A simple expression for the space charge distortion is derived which is based on the assumption that interactions produce tracks with dN/d<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">h </span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">equal to a constant and that all the tracks are straight lines radiating out from the interaction point. This is for primary ionization in the TPC volume and does not include back ground tracks or positive ion leaks from the wire chambers. A closed form expression for the distortion is obtained </span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">by using a one dimensional solution to Poisson's equation. Results are shown for the highest </span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">projected AuAu luminosities at RHIC II. This note is based on work done in 1996 to predict space charge effects that we might encounter at RHIC startup. </span></span></span></p></div><div id="mc-region-2850" style="position: absolute; top: 230.25pt; left: 6pt; width: 210.5pt; "><a name="" /><p class="Style1">The positive ion generation rate per unit volume:</p></div><div id="mc-region-2852" style="position: absolute; top: 254.25pt; left: 6pt; width: 109.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0004_17045625.PNG" id="generatedImage2852" path=".\TPC_Rev_2006AbrvForRon_images/IMG0004_17045625.PNG" style="width: 109.5pt;height: 33.75pt;"></img></div><div id="mc-region-2854" style="position: absolute; top: 266.25pt; left: 210pt; width: 177.5pt; "><a name="" /><p class="Style1">Derived in Appendix A1, this is eq. A1-8</p></div><div id="mc-region-2870" style="position: absolute; top: 296.25pt; left: 6pt; width: 437pt; "><a name="" /><p class="Style1">Note, since this is a Mathcad document it has been convenient to express variable names for derivative values as follows, for example:</p></div><div id="mc-region-2878" style="position: absolute; top: 326.25pt; left: 108pt; width: 15.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0005_17045625.PNG" id="generatedImage2878" path=".\TPC_Rev_2006AbrvForRon_images/IMG0005_17045625.PNG" style="width: 15.75pt;height: 28.5pt;"></img></div><div id="mc-region-2877" style="position: absolute; top: 332.25pt; left: 150pt; width: 53pt; "><a name="" /><p class="Style1">appears as</p></div><div id="mc-region-2876" style="position: absolute; top: 332.25pt; left: 234pt; width: 24pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0006_17045734.PNG" id="generatedImage2876" path=".\TPC_Rev_2006AbrvForRon_images/IMG0006_17045734.PNG" style="width: 24pt;height: 13.5pt;"></img></div><div id="mc-region-2475" style="position: absolute; top: 374.25pt; left: 12pt; width: 51.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0007_17045734.PNG" id="generatedImage2475" path=".\TPC_Rev_2006AbrvForRon_images/IMG0007_17045734.PNG" style="width: 51.75pt;height: 13.5pt;"></img></div><div id="mc-region-2476" style="position: absolute; top: 374.25pt; left: 102pt; width: 286.25pt; "><a name="" /><p class="Style1">Measured number from James Dunlop for min-bias AuAu with the corresponding trigger cross section for the ZDC detectors </p></div><div id="mc-region-2482" style="position: absolute; top: 404.25pt; left: 6pt; width: 48.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0008_17045750.PNG" id="generatedImage2482" path=".\TPC_Rev_2006AbrvForRon_images/IMG0008_17045750.PNG" style="width: 48.75pt;height: 13.5pt;"></img></div><div id="mc-region-2485" style="position: absolute; top: 446.25pt; left: 6pt; width: 88.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0009_17045750.PNG" id="generatedImage2485" path=".\TPC_Rev_2006AbrvForRon_images/IMG0009_17045750.PNG" style="width: 88.5pt;height: 33.75pt;"></img></div><div id="mc-region-2535" style="position: absolute; top: 452.25pt; left: 144pt; width: 200pt; "><a name="" /><p class="Style1">RHIC II predicted max luminosity with Au+Au</p></div><div id="mc-region-2569" style="position: absolute; top: 470.25pt; left: 144pt; width: 272pt; "><a name="" /><p class="Style1">http://hepwww.physics.yale.edu/star/tpc-study/Luminosity.pdf</p></div><div id="mc-region-1015" style="position: absolute; top: 500.25pt; left: 6pt; width: 58.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0010_17045750.PNG" id="generatedImage1015" path=".\TPC_Rev_2006AbrvForRon_images/IMG0010_17045750.PNG" style="width: 58.5pt;height: 27.75pt;"></img></div><div id="mc-region-1016" style="position: absolute; top: 506.25pt; left: 102pt; width: 212pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">charge density along track for minimum Ionizing</p></div><div id="mc-region-2606" style="position: absolute; top: 554.25pt; left: 108pt; width: 185pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">average charge density along track length from HIJET total charge (I. Sakrejda).</p></div><div id="mc-region-2605" style="position: absolute; top: 554.25pt; left: 6pt; width: 66pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0011_17045750.PNG" id="generatedImage2605" path=".\TPC_Rev_2006AbrvForRon_images/IMG0011_17045750.PNG" style="width: 66pt;height: 27.75pt;"></img></div><div id="mc-region-2607" style="position: absolute; top: 596.25pt; left: 24pt; width: 317pt; "><a name="" /><p class="Style1">So use the HIJET number for ionization density along the track which is 50% higher than MinI</p></div><div id="mc-region-2494" style="position: absolute; top: 648.75pt; left: 6pt; width: 108pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0013_17045750.PNG" id="generatedImage2494" path=".\TPC_Rev_2006AbrvForRon_images/IMG0013_17045750.PNG" style="width: 108pt;height: 33.75pt;"></img></div><div id="mc-region-2883" style="position: absolute; top: 656.25pt; left: 162pt; width: 234.5pt; "><a name="" /><p class="Style1">positive ion creation rate at the inner radius, note the creation rate is uniform in z independent of where the interaction occurred. This is a consequence of the uniform dN/d<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">h</span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; "> assumption.</span></span></span></p></div><div id="mc-region-2888" style="position: absolute; top: 722.25pt; left: 6pt; width: 419pt; "><a name="" /><p class="Style1">To give a quantitative feeling for this process, the positive ion creation density at the inner radius from a single average min bias AuAu event is:</p></div><div id="mc-region-575" style="position: absolute; top: 756.75pt; left: 6pt; width: 84.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0014_17045750.PNG" id="generatedImage575" path=".\TPC_Rev_2006AbrvForRon_images/IMG0014_17045750.PNG" style="width: 84.75pt;height: 33.75pt;"></img></div><div id="mc-region-2886" style="position: absolute; top: 812.25pt; left: 6pt; width: 429.5pt; "><a name="" /><p class="Style1">The positive ions created through out the TPC volume drift to the central membrane where they are collected. The steady state positive ion density is obtained by integrating the positive ion source rate over z and by knowing the positive ion drift velocity. </p></div><div id="mc-region-577" style="position: absolute; top: 865.5pt; left: 6pt; width: 63.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0015_17045750.PNG" id="generatedImage577" path=".\TPC_Rev_2006AbrvForRon_images/IMG0015_17045750.PNG" style="width: 63.75pt;height: 12.75pt;"></img></div><div id="mc-region-2506" style="position: absolute; top: 864.75pt; left: 108pt; width: 82.25pt; "><a name="" /><p class="Style1">drift length of TPC</p></div><div id="mc-region-578" style="position: absolute; top: 900.75pt; left: 96pt; width: 332pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">positive ion drift velocity in the z direction, this comes from Fig. 2.5, p. 63 in Blume and Rolandi. It is the positive ion drift velocity in P10 at 130 V/cm</p></div><div id="mc-region-579" style="position: absolute; top: 900.75pt; left: 12pt; width: 53.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0016_17045765.PNG" id="generatedImage579" path=".\TPC_Rev_2006AbrvForRon_images/IMG0016_17045765.PNG" style="width: 53.25pt;height: 27.75pt;"></img></div><div id="mc-region-580" style="position: absolute; top: 948.75pt; left: 6pt; width: 434pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The charge density at points in the TPC caused by positive ions flowing from their points of creation to the central membrane is given as:</p></div><div id="mc-region-581" style="position: absolute; top: 981.75pt; left: 6pt; width: 276pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0017_17045765.PNG" id="generatedImage581" path=".\TPC_Rev_2006AbrvForRon_images/IMG0017_17045765.PNG" style="width: 276pt;height: 48pt;"></img></div><div id="mc-region-2507" style="position: absolute; top: 1044.75pt; left: 6pt; width: 194.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0018_17045765.PNG" id="generatedImage2507" path=".\TPC_Rev_2006AbrvForRon_images/IMG0018_17045765.PNG" style="width: 194.25pt;height: 33.75pt;"></img></div><div id="mc-region-2573" style="position: absolute; top: 1050.75pt; left: 258pt; width: 176.75pt; "><a name="" /><p class="Style1">steady state positive ion density (space charge)</p></div><div id="mc-region-2470" style="position: absolute; top: 1104.75pt; left: 6pt; width: 138pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0019_17045765.PNG" id="generatedImage2470" path=".\TPC_Rev_2006AbrvForRon_images/IMG0019_17045765.PNG" style="width: 138pt;height: 33.75pt;"></img></div><div id="mc-region-2510" style="position: absolute; top: 1152.75pt; left: 6pt; width: 123.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0020_17045765.PNG" id="generatedImage2510" path=".\TPC_Rev_2006AbrvForRon_images/IMG0020_17045765.PNG" style="width: 123.75pt;height: 33.75pt;"></img></div><div id="mc-region-2604" style="position: absolute; top: 1188.75pt; left: 6pt; width: 437pt; "><a name="" /><p class="Style1">With the much higher luminosity expected for RHIC II this positive ion density is 200 times what was predicted originally in Gulshan Rai's STAR note #3 </p></div><div id="mc-region-2512" style="position: absolute; top: 1248.75pt; left: 6pt; width: 437pt; "><a name="" /><p class="Style1">In the following an approximation of the distortion is obtained from this space charge density by using a one dimensional solution to Poison's equation. The approximation over predicts the distortion and is ~50% larger than a full 3D Poison solution. </p></div><div id="mc-region-101" style="position: absolute; top: 1320.75pt; left: 6pt; width: 437pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The radial E field (Er) can be approximated using Gausses law if we treat the TPC tube as <span style="font-family: 'Symbol', sans-serif;">¥</span> length with a uniform charge density along the tube. Then:</p></div><div id="mc-region-111" style="position: absolute; top: 1366.5pt; left: 6pt; width: 157.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0021_17045765.PNG" id="generatedImage111" path=".\TPC_Rev_2006AbrvForRon_images/IMG0021_17045765.PNG" style="width: 157.5pt;height: 41.25pt;"></img></div><div id="mc-region-114" style="position: absolute; top: 1434.75pt; left: 6pt; width: 441.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The integration surface is the inner field cage metal where the field is 0 and the outer cylinder is set at radius r where we wish to evaluate the radial field. The charge included in the volume of integration is the space charge derived above plus an induced surface charge, <span style="font-family: 'Symbol', sans-serif;">s</span><sub>in</sub> ,on the inner field cage.</p></div><div id="mc-region-116" style="position: absolute; top: 1482.75pt; left: 12pt; width: 140pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The integrals above reduce to:</p></div><div id="mc-region-115" style="position: absolute; top: 1512pt; left: 6pt; width: 256.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0022_17045828.PNG" id="generatedImage115" path=".\TPC_Rev_2006AbrvForRon_images/IMG0022_17045828.PNG" style="width: 256.5pt;height: 41.25pt;"></img></div><div id="mc-region-451" style="position: absolute; top: 1587.75pt; left: 12pt; width: 383.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0023_17045828.PNG" id="generatedImage451" path=".\TPC_Rev_2006AbrvForRon_images/IMG0023_17045828.PNG" style="width: 383.25pt;height: 56.25pt;"></img></div><div id="mc-region-457" style="position: absolute; top: 1653.75pt; left: 12pt; width: 295.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0024_17045828.PNG" id="generatedImage457" path=".\TPC_Rev_2006AbrvForRon_images/IMG0024_17045828.PNG" style="width: 295.5pt;height: 50.25pt;"></img></div><div id="mc-region-583" style="position: absolute; top: 1726.5pt; left: 6pt; width: 280.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0025_17045828.PNG" id="generatedImage583" path=".\TPC_Rev_2006AbrvForRon_images/IMG0025_17045828.PNG" style="width: 280.5pt;height: 33pt;"></img></div><div id="mc-region-2890" style="position: absolute; top: 1772.25pt; left: 6pt; width: 427.25pt; "><a name="" /><p class="Style1">Keep in mind the confusing variable names, namely <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">s</span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; "> is the average min bias AuAu cross section and </span></span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="color: #00f; "><span style="font-weight: normal; font-style: normal;">s</span></span></span><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; "><sub><span style="font-size: 10pt; font-family: 'Arial', sans-serif;">in</span></sub></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; "> is the surface charge density on the inner field cage induced by the space charge.</span></span></span></p></div><div id="mc-region-119" style="position: absolute; top: 1830.75pt; left: 12pt; width: 429.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The surface charge, <span style="font-family: 'Symbol', sans-serif;">s</span><sub>in</sub> , is obtained knowing that the potential between the inner and outer field cage is 0 at any given z.</p></div><div id="mc-region-157" style="position: absolute; top: 1881.75pt; left: 6pt; width: 405.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0026_17045843.PNG" id="generatedImage157" path=".\TPC_Rev_2006AbrvForRon_images/IMG0026_17045843.PNG" style="width: 405.75pt;height: 53.25pt;"></img></div><div id="mc-region-467" style="position: absolute; top: 1971.75pt; left: 6pt; width: 329.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0027_17045843.PNG" id="generatedImage467" path=".\TPC_Rev_2006AbrvForRon_images/IMG0027_17045843.PNG" style="width: 329.25pt;height: 43.5pt;"></img></div><div id="mc-region-477" style="position: absolute; top: 2063.25pt; left: 6pt; width: 192pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0028_17045843.PNG" id="generatedImage477" path=".\TPC_Rev_2006AbrvForRon_images/IMG0028_17045843.PNG" style="width: 192pt;height: 50.25pt;"></img></div><div id="mc-region-476" style="position: absolute; top: 2164.5pt; left: 6pt; width: 280.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0029_17045843.PNG" id="generatedImage476" path=".\TPC_Rev_2006AbrvForRon_images/IMG0029_17045843.PNG" style="width: 280.5pt;height: 33pt;"></img></div><div id="mc-region-584" style="position: absolute; top: 2226.75pt; left: 12pt; width: 83pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">substituting for<span style="font-family: 'Symbol', sans-serif;"> </span><span style="font-family: 'Symbol', sans-serif;">s</span><sub>in</sub></p></div><div id="mc-region-2893" style="position: absolute; top: 2217pt; left: 6pt; width: 216.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0030_17045843.PNG" id="generatedImage2893" path=".\TPC_Rev_2006AbrvForRon_images/IMG0030_17045843.PNG" style="width: 216.75pt;height: 54pt;"></img></div><div id="mc-region-2894" style="position: absolute; top: 2246.25pt; left: 264pt; width: 166.25pt; "><a name="" /><p class="Style1">finally the radial component of the electric field caused by space charge</p></div><div id="mc-region-2586" style="position: absolute; top: 2334.75pt; left: 6pt; width: 418.25pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">The distortion in the r<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">f</span></span></span> direction comes from the <span style="font-family: 'Arial', sans-serif;">EXB term integrated along the drift path</span> and is:</p></div><div id="mc-region-1287" style="position: absolute; top: 2371.5pt; left: 6pt; width: 428.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0031_17045843.PNG" id="generatedImage1287" path=".\TPC_Rev_2006AbrvForRon_images/IMG0031_17045843.PNG" style="width: 428.25pt;height: 62.25pt;"></img></div><div id="mc-region-1294" style="position: absolute; top: 2461.5pt; left: 6pt; width: 251.25pt; background-color: inherit; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0032_17045859.PNG" id="generatedImage1294" path=".\TPC_Rev_2006AbrvForRon_images/IMG0032_17045859.PNG" style="width: 251.25pt;height: 57.75pt;"></img></div><div id="mc-region-2513" style="position: absolute; top: 2490.75pt; left: 306pt; width: 137pt; "><a name="" /><p class="Style1">distortion in the r<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">f</span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; "> direction</span></span></span></p></div><div id="mc-region-2592" style="position: absolute; top: 2550.75pt; left: 6pt; width: 53.75pt; "><a name="" /><p class="Style1">assume <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">w</span></span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">t</span></span></span></p></div><div id="mc-region-2593" style="position: absolute; top: 2574.75pt; left: 6pt; width: 41.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0033_17045906.PNG" id="generatedImage2593" path=".\TPC_Rev_2006AbrvForRon_images/IMG0033_17045906.PNG" style="width: 41.25pt;height: 13.5pt;"></img></div><div id="mc-region-2900" style="position: absolute; top: 2576.25pt; left: 120pt; width: 233.75pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">for 0.5 Tesla in P10, this may not be the latest value</p></div><div id="mc-region-2587" style="position: absolute; top: 2611.5pt; left: 6pt; width: 48pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0034_17045906.PNG" id="generatedImage2587" path=".\TPC_Rev_2006AbrvForRon_images/IMG0034_17045906.PNG" style="width: 48pt;height: 15pt;"></img></div><div id="mc-region-2899" style="position: absolute; top: 2612.25pt; left: 120pt; width: 99.5pt; "><a name="" /><p class="Style1">inner field cage radius</p></div><div id="mc-region-2588" style="position: absolute; top: 2635.5pt; left: 6pt; width: 56.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0035_17045906.PNG" id="generatedImage2588" path=".\TPC_Rev_2006AbrvForRon_images/IMG0035_17045906.PNG" style="width: 56.25pt;height: 15pt;"></img></div><div id="mc-region-2898" style="position: absolute; top: 2636.25pt; left: 120pt; width: 100.25pt; "><a name="" /><p class="Style1">outer field cage radius</p></div><div id="mc-region-1344" style="position: absolute; top: 2664.75pt; left: 6pt; width: 60pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0036_17045906.PNG" id="generatedImage1344" path=".\TPC_Rev_2006AbrvForRon_images/IMG0036_17045906.PNG" style="width: 60pt;height: 27.75pt;"></img></div><div id="mc-region-2897" style="position: absolute; top: 2672.25pt; left: 120pt; width: 41pt; "><a name="" /><p class="Style1">drift field</p></div><div id="mc-region-2619" style="position: absolute; top: 2720.25pt; left: 6pt; width: 321pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0039_17045906.PNG" id="generatedImage2619" path=".\TPC_Rev_2006AbrvForRon_images/IMG0039_17045906.PNG" style="width: 321pt;height: 293.25pt;"></img></div><div id="mc-region-2621" style="position: absolute; top: 3046.5pt; left: 18pt; width: 403.25pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-indent: 0pt; text-align: left; list-style-type: none;">Fig. 1 r<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">f</span></span></span> distortion for a straight line track. A straight line vertical track at the center of TPC (dotted line) will be distorted by space charge and appears as the solid line.</p></div><div id="mc-region-2539" style="position: absolute; top: 3188.25pt; left: 12pt; width: 402pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0042_17045921.PNG" id="generatedImage2539" path=".\TPC_Rev_2006AbrvForRon_images/IMG0042_17045921.PNG" style="width: 402pt;height: 389.25pt;"></img></div><div id="mc-region-2618" style="position: absolute; top: 3598.5pt; left: 18pt; width: 389pt; "><a name="" /><p class="Style1">Fig. 2 Steady state positive ion density as a function of r and z in the TPC volume for RHIC II predicted AuAu luminosity. This is the positive ion density that produced the distortion shown in Fig. 1 </p></div><div id="mc-region-2678" style="position: absolute; top: 3655.5pt; left: 6pt; width: 71.75pt; "><a name="" /><p class="Style1"><span style="font-size: 13.26315789473684pt; ">Appendix A1</span></p></div><div id="mc-region-2679" style="position: absolute; top: 3688.5pt; left: 6pt; width: 57.5pt; "><a name="" /><p class="Style1">Derivation of</p></div><div id="mc-region-2680" style="position: absolute; top: 3722.25pt; left: 6pt; width: 147pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0043_17045921.PNG" id="generatedImage2680" path=".\TPC_Rev_2006AbrvForRon_images/IMG0043_17045921.PNG" style="width: 147pt;height: 36pt;"></img></div><div id="mc-region-2705" style="position: absolute; top: 3736.5pt; left: 198pt; width: 238.25pt; "><a name="" /><p class="Style1">Eq. A1-1 the rate of positive ion generation per volume</p></div><div id="mc-region-2895" style="position: absolute; top: 3776.25pt; left: 6pt; width: 221pt; "><a name="" /><p class="Style1">using a cylindrical coordinate geometry (Fig. A1-1).</p></div><div id="mc-region-2807" style="position: absolute; top: 3814.5pt; left: 18pt; width: 15.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0044_17045921.PNG" id="generatedImage2807" path=".\TPC_Rev_2006AbrvForRon_images/IMG0044_17045921.PNG" style="width: 15.75pt;height: 28.5pt;"></img></div><div id="mc-region-2808" style="position: absolute; top: 3820.5pt; left: 66pt; width: 268.25pt; "><a name="" /><p class="Style1">number of tracks per unit pseudo rapidity, assumed constant</p></div><div id="mc-region-2809" style="position: absolute; top: 3848.25pt; left: 18pt; width: 18pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0045_17045921.PNG" id="generatedImage2809" path=".\TPC_Rev_2006AbrvForRon_images/IMG0045_17045921.PNG" style="width: 18pt;height: 30pt;"></img></div><div id="mc-region-2810" style="position: absolute; top: 3856.5pt; left: 66pt; width: 295.25pt; "><a name="" /><p class="Style1">number of ion-electron pairs created per unit length along the track </p></div><div id="mc-region-2799" style="position: absolute; top: 3899.25pt; left: 18pt; width: 6pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0046_17045937.PNG" id="generatedImage2799" path=".\TPC_Rev_2006AbrvForRon_images/IMG0046_17045937.PNG" style="width: 6pt;height: 12.75pt;"></img></div><div id="mc-region-2817" style="position: absolute; top: 3898.5pt; left: 66pt; width: 52.25pt; "><a name="" /><p class="Style1">Luminosity</p></div><div id="mc-region-2811" style="position: absolute; top: 3934.5pt; left: 18pt; width: 6.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0047_17045937.PNG" id="generatedImage2811" path=".\TPC_Rev_2006AbrvForRon_images/IMG0047_17045937.PNG" style="width: 6.75pt;height: 13.5pt;"></img></div><div id="mc-region-2816" style="position: absolute; top: 3934.5pt; left: 66pt; width: 63.5pt; "><a name="" /><p class="Style1">cross section</p></div><div id="mc-region-2814" style="position: absolute; top: 3959.25pt; left: 18pt; width: 9pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0048_17045937.PNG" id="generatedImage2814" path=".\TPC_Rev_2006AbrvForRon_images/IMG0048_17045937.PNG" style="width: 9pt;height: 15pt;"></img></div><div id="mc-region-2815" style="position: absolute; top: 3958.5pt; left: 66pt; width: 71.75pt; "><a name="" /><p class="Style1">electron charge</p></div><div id="mc-region-2696" style="position: absolute; top: 3992.25pt; left: 12pt; width: 246pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0050_17045968.PNG" id="generatedImage2696" path=".\TPC_Rev_2006AbrvForRon_images/IMG0050_17045968.PNG" style="width: 246pt;height: 303.75pt;"></img></div><div id="mc-region-2672" style="position: absolute; top: 4318.5pt; left: 30pt; width: 333.5pt; "><a name="" /><p class="Style1">Fig. A1-1 Elemental volume with track along s leaving ionization segment <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">d</span></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><span style="color: #00f; ">s</span></span></span></p></div><div id="mc-region-2702" style="position: absolute; top: 4348.5pt; left: 12pt; width: 182pt; "><a name="" /><p class="Style1">from the geometry illustrated in Fig. A1-1</p></div><div id="mc-region-2703" style="position: absolute; top: 4370.25pt; left: 12pt; width: 99.75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0051_17045984.PNG" id="generatedImage2703" path=".\TPC_Rev_2006AbrvForRon_images/IMG0051_17045984.PNG" style="width: 99.75pt;height: 30pt;"></img></div><div id="mc-region-2706" style="position: absolute; top: 4378.5pt; left: 150pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-2</p></div><div id="mc-region-2708" style="position: absolute; top: 4408.5pt; left: 18pt; width: 30.5pt; "><a name="" /><p class="Style1">where</p></div><div id="mc-region-2714" style="position: absolute; top: 4426.5pt; left: 234pt; width: 199.25pt; "><a name="" /><p class="Style1">is for a collision, the average number of tracks per area impinging on the inside surface of the elemental volume in Fig. A1-1 </p></div><div id="mc-region-2709" style="position: absolute; top: 4425.75pt; left: 12pt; width: 140.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0052_17045984.PNG" id="generatedImage2709" path=".\TPC_Rev_2006AbrvForRon_images/IMG0052_17045984.PNG" style="width: 140.25pt;height: 29.25pt;"></img></div><div id="mc-region-2717" style="position: absolute; top: 4432.5pt; left: 174pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-3</p></div><div id="mc-region-2821" style="position: absolute; top: 4480.5pt; left: 12pt; width: 23.75pt; "><a name="" /><p class="Style1">and </p></div><div id="mc-region-2820" style="position: absolute; top: 4467pt; left: 54pt; width: 104.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0053_17045984.PNG" id="generatedImage2820" path=".\TPC_Rev_2006AbrvForRon_images/IMG0053_17045984.PNG" style="width: 104.25pt;height: 35.25pt;"></img></div><div id="mc-region-2824" style="position: absolute; top: 4480.5pt; left: 198pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-4</p></div><div id="mc-region-2828" style="position: absolute; top: 4480.5pt; left: 264pt; width: 170pt; "><a name="" /><p class="Style1">is the length of the track segment passing through the elemental volume</p></div><div id="mc-region-2831" style="position: absolute; top: 4516.5pt; left: 12pt; width: 51pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0054_17045984.PNG" id="generatedImage2831" path=".\TPC_Rev_2006AbrvForRon_images/IMG0054_17045984.PNG" style="width: 51pt;height: 13.5pt;"></img></div><div id="mc-region-2832" style="position: absolute; top: 4516.5pt; left: 198pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-5</p></div><div id="mc-region-2830" style="position: absolute; top: 4516.5pt; left: 264pt; width: 107pt; "><a name="" /><p class="Style1">is the elemental volume</p></div><div id="mc-region-2738" style="position: absolute; top: 4558.5pt; left: 6pt; width: 194.75pt; "><a name="" /><p class="Style1">combining Eq. A1-2 through Eq. A1-5 gives</p></div><div id="mc-region-2744" style="position: absolute; top: 4589.25pt; left: 6pt; width: 334.5pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0055_17045984.PNG" id="generatedImage2744" path=".\TPC_Rev_2006AbrvForRon_images/IMG0055_17045984.PNG" style="width: 334.5pt;height: 57pt;"></img></div><div id="mc-region-2746" style="position: absolute; top: 4618.5pt; left: 366pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-6</p></div><div id="mc-region-2749" style="position: absolute; top: 4672.5pt; left: 12pt; width: 222.5pt; "><a name="" /><p class="Style1">the charge volume density left by a single collision</p></div><div id="mc-region-2751" style="position: absolute; top: 4714.5pt; left: 18pt; width: 36.5pt; "><a name="" /><p class="Style1">To find </p></div><div id="mc-region-2752" style="position: absolute; top: 4707.75pt; left: 90pt; width: 15pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0056_17045984.PNG" id="generatedImage2752" path=".\TPC_Rev_2006AbrvForRon_images/IMG0056_17045984.PNG" style="width: 15pt;height: 28.5pt;"></img></div><div id="mc-region-2754" style="position: absolute; top: 4714.5pt; left: 156pt; width: 161pt; "><a name="" /><p class="Style1">use the definition of pseudo rapidity </p></div><div id="mc-region-2758" style="position: absolute; top: 4755.75pt; left: 12pt; width: 75pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0057_17046000.PNG" id="generatedImage2758" path=".\TPC_Rev_2006AbrvForRon_images/IMG0057_17046000.PNG" style="width: 75pt;height: 28.5pt;"></img></div><div id="mc-region-2760" style="position: absolute; top: 4762.5pt; left: 120pt; width: 23.75pt; "><a name="" /><p class="Style1">and </p></div><div id="mc-region-2761" style="position: absolute; top: 4756.5pt; left: 180pt; width: 53.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0058_17046000.PNG" id="generatedImage2761" path=".\TPC_Rev_2006AbrvForRon_images/IMG0058_17046000.PNG" style="width: 53.25pt;height: 27.75pt;"></img></div><div id="mc-region-2763" style="position: absolute; top: 4792.5pt; left: 12pt; width: 29.75pt; "><a name="" /><p class="Style1">to get</p></div><div id="mc-region-2765" style="position: absolute; top: 4821pt; left: 6pt; width: 111pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0059_17046000.PNG" id="generatedImage2765" path=".\TPC_Rev_2006AbrvForRon_images/IMG0059_17046000.PNG" style="width: 111pt;height: 35.25pt;"></img></div><div id="mc-region-2767" style="position: absolute; top: 4870.5pt; left: 6pt; width: 108.5pt; "><a name="" /><p class="Style1">and taking the derivative</p></div><div id="mc-region-2769" style="position: absolute; top: 4899.75pt; left: 6pt; width: 65.25pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0060_17046000.PNG" id="generatedImage2769" path=".\TPC_Rev_2006AbrvForRon_images/IMG0060_17046000.PNG" style="width: 65.25pt;height: 36pt;"></img></div><div id="mc-region-2772" style="position: absolute; top: 4912.5pt; left: 108pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-7</p></div><div id="mc-region-2792" style="position: absolute; top: 4954.5pt; left: 18pt; width: 178.25pt; "><a name="" /><p class="Style1">combine Eq. A1-7 and A1-6 to eliminate</p></div><div id="mc-region-2793" style="position: absolute; top: 4947.75pt; left: 210pt; width: 15pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0061_17046000.PNG" id="generatedImage2793" path=".\TPC_Rev_2006AbrvForRon_images/IMG0061_17046000.PNG" style="width: 15pt;height: 28.5pt;"></img></div><div id="mc-region-2865" style="position: absolute; top: 4994.25pt; left: 6pt; width: 132.5pt; "><a name="" /><p class="Style1">and multiply by the event rate</p></div><div id="mc-region-2866" style="position: absolute; top: 4994.25pt; left: 192pt; width: 15pt; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0062_17046062.PNG" id="generatedImage2866" path=".\TPC_Rev_2006AbrvForRon_images/IMG0062_17046062.PNG" style="width: 15pt;height: 13.5pt;"></img></div><div id="mc-region-2864" style="position: absolute; top: 5024.25pt; left: 6pt; width: 307.25pt; "><a name="" /><p class="Style1">gives the following expression for the generation rate of positive ions per volume</p></div><div id="mc-region-2795" style="position: absolute; top: 5066.25pt; left: 18pt; width: 118.5pt; background-color: inherit; "><a name="" /><img border="0" src=".\TPC_Rev_2006AbrvForRon_images/IMG0063_17046062.PNG" id="generatedImage2795" path=".\TPC_Rev_2006AbrvForRon_images/IMG0063_17046062.PNG" style="width: 118.5pt;height: 36pt;"></img></div><div id="mc-region-2796" style="position: absolute; top: 5074.5pt; left: 162pt; width: 43.25pt; "><a name="" /><p class="Style1">Eq. A1-8</p></div><div id="mc-region-2794" style="position: absolute; top: 5074.5pt; left: 234pt; width: 202.25pt; "><a name="" /><p class="Style1">which is uniform in z, i.e. it does not matter where in z the collision occurred.</p></div><div id="mc-region-2862" style="position: absolute; top: 5126.25pt; left: 18pt; width: 425pt; "><a name="" /><p class="Style1">A similar and perhaps more straight forward derivation has been done starting with spherical coordinates which gives the same expression. This will be added later.</p></div><div id="mc-region-2610" style="position: absolute; top: 5182.5pt; left: 30pt; width: 383.75pt; "><a name="" /><p class="Style1">References:</p><p class="Style1"></p><p class="Style1">1. C:\Documents and Settings\Howard Wieman\My Documents\space charge distortion STAR\ TwoD5.MCD</p></div><div></div></body></html>